The following is a presentation of TFNN The Traders Edge with Steve Rhodes Toll free at 1-877-927-6648 or internationally at<PHONE_NUMBER> The Traders Edge now Steve Rhodes I've always basled Chapman sitting for Steve Rhodes sitting in for the hour. No way I could do the Traders Edge on there That's the unique thing about all the hosts here TFNN. We all have our individual techniques Proprietary to techniques and actually techniques we discuss in public We have webinars on all these different techniques, etc. And I'm just thinking as I say that our late Dave White colleague Evidently would have been his birthday on the 29th leap year So we miss you Dave and Deepest thoughts we miss you a lot here So with that I'm going to do this. So the Dow right now is up 61 at 39,058 We it might work in the Chapman Wave. We've got a very simple technique It does get complex as it moves on but the simplicity is you identify a low And you can't eat success of the higher peak of appetizing sequentially uppercase on the way up lower case on the way down ABCDEFG there's never an H and D other things can happen if a D within three bars It makes a new high you call that an alternate count E and then slash a f slash B g src And invariably she says she goes to the D might pull back first, but it'll go to the D Simple bicep don't get upgraded to a by mode The objective is to always get you to at least a leg D and then a peak D It can go do all sorts of things at D, but that's the objective. Well, what do we do? We got to a D right here in the Dady chart pullback We've created an up channel Chapman Wave inside track up channel support zone We've gone right to the Chapman Wave inside track repellent zone and got repelled. It's holding the nine the black nine period moving average I like to use one. I called the indicator of last resort because not this one right here Why because when the nine period moving average crosses positive and look here in the spine cross positive on March I think sorry November the third Even with all the stuff that's gone on in the news or whatever whatever whatever It has not crossed there even with that sharp pullback early in January it did not cross negative And it's still the nine period moving average If you just held this one position And use that you would still be long we've tried to do that with Microsoft. We've had it for quite some time We've had it since there And look it's holding the we even it looked like it was gonna go pink and didn't go pink It could still go pink, but this is an indicator called the indicator of last resort So I'm just showing you these things to show you how Some simple techniques can work and you can make them complex I sometimes make it complex by trying to identify a very short term tough We've actually we are still on a very short term basis Short the Dow that via the DOG. We still have long positions from March of 2020 In the diamonds and the diamond and UDOW should never do that But we do have it three times long from October of 2022 Yeah, so it's just a short term trading position But we'll see if it's going to work now. Let me show you a couple of things and I want to go I've got all these questions. I say that I will try to do what Steve there so well, I'll try to take as many questions as possible So this is the S&P leg G slash C right now going towards the inside track repellent zone. Yes It made a new high today. That means the weekly chart is only a leg B After this fulcrum this propeller shaft moved to the outside So any pullback makes this whole area no matter how deep the whole area of 4,800 to 4,700 should hold If there's a really sharp and it's still in a buy mode. So this is very positive looking out and Looking at the QQQ I want to get this done very quickly because I've got now got a ton of questions that come in A leg D in the daily chart of 443.12 up 413 And it's at an all-time high as we speak and there's only a leg C in the weekly chart So to get it to a D you've got to go out at least another minimum of three weeks All right That means every week you either makes a lower high next week and then a high high the following week And then a lower high and you've made your peak D. It doesn't quite work like that in real life They take a little bit longer. So this is very positive. IWM is breaking out Of the resistance that it had in the 206 area No, sorry 205 area now it's a 206 13 Up 2.42 nice action. I like that action is very good and a leg D in the weekly chart And only a leg B and you and I have to call it I was calling it a gray leg B now I can call it a blue meaning that it's a buy mode In the monthly chart for the first time it's done there. Oh in ages Look the magnies go the stochastics finally at 81 percent and we still that means we closed last month yesterday In the 80 81 percent area now you had 81 starting March And the on balance volume is not overboard at all. It's kind of just okay And the nine is crossed positive. So this is looking very good. I didn't do the XLF I was asked about and I'll do the XLF That's the financials starting here to peak air from the daily chart Look way above the nine or not away, but it's above the nine, but the nine is way above the 14 Magnies good not great, but good stochastic is very good at 90 percent on balance volume It's a little bit overboard weekly chart is energy slash C I'm looking at this and the monthly chart has to get to 41 70 January high 2022 after falling to 29 In October of 2022 very nice candle very nice candle last month and because we've gone Oh, we have we must go to a higher high this month above the high that was made last month of last week of 40.55 40.56 Continues the monthly leg b and continues leg g says c in the weekly chart. Uh now I want to go to gold Gold is broken out very nicely. I was talking about this in the show I say the 2082 level on the left with a nice price time match left side right side Couple v shape formation might challenge that very quickly. I'm wrong. Behold. We've just done that up 27 dollars The weekly chart. Hey, it's it's not bad. It's good It's a nice start But look the nine-period moving average with all of this pullback from the two Um from the 2100s down to just under 2000 that nine-period moving average held So this is a very this is a big positive So, um, yeah, I just want to finish that up with Dollar just pulling back a little bit down 21 ticks at 103.93 That high that was made uh, just uh three weeks ago At 105 just I think it was 105 104.98 That's the only nicely. I don't bet the nine The the nine-period moving average pink then it went green yesterday. Now it's pink again when you get this You've got to be careful that it isn't producing a magnet line at the 200 period moving average of 103.80 If you already want to be bearish now on the dollar and bullish and gold You want to see this down as 102.30 by this coming friday or at least Maybe seven sessions from now trading under that that I'll say okay now gold can be higher I'll be back 1040 sbs of 20 and I've got a bunch of calls. We'll go to a net very soon that's uh Arista networks If you're looking for potential trading setups in the stock market then rocket equities and options report is a newsletter You should try tamio brian delivers options and equity trades when the markets present them Using a combination of fundamentals and technicals sign up for rocket equities and options report today with a 30 day money back guarantee So you have nothing to risk for all the details and to start your subscription today Visit the front page of tfnn.com tfnn educating investors You might think that if you want to be successful at trading in the stock market You're going to need a crystal ball after all it's impossible to predict the future right like any endeavor in life Before you decide it's impossible Get some advice from the experts you might find that it's not so impossible after all for daily market overviews That give you direction on the key indices selective stocks and commodities Subscribe to the opening call newsletter at tfnn.com The opening call newsletter is written by basal chapman creator of the trading methodology known as the chapman wave The chapman wave up down sequence gives you an edge in identifying price turns finding the peaks and valleys and stock prices Get the opening call newsletter by basal chapman and your inbox every day first time subscribers also get a 30 day money back guarantee If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up tfnn.com educating investors Steve road started his trading career as a student almost 20 years ago and the student has now become the master Steve won the prestigious timer of the year award in 2018 and barely missed that mark again in 2019 Finishing at number two for the year an amazing accomplishment Steve Rhodes is committed to sharing his techniques and knowledge with anyone who wants to learn And he shares his vast amount of trading knowledge every day in his mastering probability newsletter Steve's award-winning newsletter Mastering probability is delivered every trading day with updates throughout the afternoon Sign up for steve's market newsletter Mastering probability and you'll receive access to seven of steve's educational webinars Absolutely free at tfnn all our newsletters come with a 30 day money back guarantee So you have absolutely nothing to worry about visit tfnn.com and try mastering probability 30 days risk-free today tfnn educating investors tfnn has launched the tiger's den hosted at discord tfnn has been educating traders for more than 20 years with live programming hosted by a variety of professional traders during market hours The tiger's den available to all tigers and tigeresses for just one dollar for the year There's no cash or added costs when you join our community of traders sign up today and become a part of this educational community of traders Just visit the front page of tfnn.com Toll free at 1-877-927-6648 internationally at<PHONE_NUMBER> The tiger, you're a dazzle chap, I'm staying in from Seabro's hour In the uh in the tiger youtuber someone said uh who was it? Jeff? Was it uh Have a look at MU Yes, micron monthly going back to 1999. I remember this so well. I I should have Charts all notated with the chap wave no notation if I hadn't lost them through different things because I've noted look at this It made a lot. I I've spoken about this I even had to I put in a webinar once where I spoke about advanced micro devices that used to go to the single digits And then scream to the hundreds or even higher than that once upon a time Look chap wave a low right there in 1996 goes to a peak deep pulls back starts over again at um In june of 1998 at 10.03. I think it was peak a peak b peak c peak d Actually a chap we've instantly stopped as it goes to e Goes to f and the month of july of 2000 makes At top at 97.50 But look at this it had a you you talk about round numbers. I remember this well it had a 91 round number Open that month Yeah, I think it was split, but it was still a round number Isn't that interesting? That's why i'm saying these round numbers are going to turn into something important I don't know yet exactly how it's going to work out then back in 09 it goes down from from those hundreds it goes down to $1.59 source of brand new peak abcd pulls back 2010 Yeah, 2011 pulls back source another one That goes to that one went to 11 95 this one goes down to 397 And the screws peak abcd e f and even a g Right there G and oh my god, I can't believe it the g in the month of december of 2014 And you know what it makes a 36.59 high But it opened the month to open the month. It's just coincidence, but it's amazing how it happens It had a round number 36. Well, it goes from 36 down to nine dollars and 31 cents A year later less than a year later January of 2016 Then it goes peak abc de Fails right there This is a chapter we've inverted green roman candle And it takes another one right there and then it turns around and it fails completely with another one And what was that high that I was 64.46 around numbers you can't expect every time Yeah, and then it pulls back from that in the 60s it goes down to get cut in half and 2018 a january no december of 2018 it goes to 28 39 goes peak abcd And then continues higher to e and f with a double top of 96.95 Back in about april of 2021 and then January it goes to 98.45 after dropping 30 points to 965 it goes to 98 double top right there Yeah, that's it and then it comes down And now we've started a brand new move to the upside From the low 48.43 and it's trading in a leg c right now. So that's still very bullish unless you make a triple top 95 right now the highest 95 97 So 98.45 was the and this is an inverted I've done a lot of work on this inverted handle and cup this is not a cup and handle handle and cup I I don't want to go into it right now because I go to review what I used to do when I did this particular pattern Yeah, it's not an n. That's a b. This is a new leg b. I hope that helps you. Okay. Next question was Arista Let's go to a net A and e t So this is having a big move today It's up 3.3 9% up 940 at 286.94 had a high of 292.60 62 had a round a couple of round quite a few round numbers over the last Three weeks. I didn't actually check to see in the most recent ones. I'm not going to waste time on that right now Oh tooth Opened yesterday 275 Had a high of 278 and yeah, it is a 287 So it's it's looking out over a period of time. It's not immediately that you look at it I want to see what happens. Will it retest the 292.66? Well, it's acting very nicely And the monthly the weekly charge is still strong in the monthly chart is amazing But when these take a breather, I think there's it's going to take quite a bit of a breather It's looking really good right now. So where's the support to 74 to 271 The 14 period and the 9 period moving averages. That's your support breaks under in the next two weeks if it closes any day Close under 267 I say just start being careful because it could be retesting the low that was made just the other day And look the pink the 9 period moving average when pink just briefly announced back to green So think of it green is positive. That's all you have to do. Now. I need to go to Um, okay, so I'm going to go to this one. Oh, so that's so where could it go to, you know for me I do this all the time and then I find the majority of time for I love the way Larry does it for some reason Larry has a way of looking at these One-to-one expansions and he does it so beautifully. I'm picking this from here To there straight line and then I want to I like the parallel moves. I like to see like A move that does one-to-one in the same angle the same degree. So if you do it this way, that's a failure Because I always take it from the low Right, but if you take it from the high We've already broken above it. So all I can say is that's not working So I can't really tell you what I would do is I'd go to my automated Uh, where did it go? All right there. So a a net a well first of all Let's do this semiconductor to two one point sixty one as a chapter wave Automated resistance level in the daily. There's nothing here in the week There's nothing in the monthly nothing in 120 millichart. So to 21 would be resistance We have a couple of points away from that. Let's go to a net and a net There's 292 point 40 and it's already gone there. It's gone to two 292 point 50 66 Every couple of weeks ago. So there's your resistance and you're breaking out in the weekly chart above the resistance That has now become support and there's nothing in the weekly of the 120-minute chart So I hope that helped you next question. I wrote them down in order Fizz fizz is Phi serve I believe Phi serves right here fis Yeah, so it made a new Recovery high today We went to uh, yes. Yes today 69 points 59 is down 34 cents. It's in a leg e The on balance volume is already pulling back. It's a little more than 90 percent In the stochastic is really good Magdy's good.  It can go a little higher towards 70 But this is what I would be looking at First of all, I'd look at this as your your first Area of resistance and support and that takes you down to the 90. Oops. I didn't mean 90. I mean 60 66 to 65 area is key support. Very strong leg d in the weekly chart Yeah, so the midpoint of that would be your first really good support But that monthly chart is only in a leg. Hey fidelity national information payment. Oh, that's what it is Fidelity national information payment services credit debit card processing. Oh, I looked at this level now. I forgot about that Yeah, acting very well I'll be back guys up 28 as the fees of 20 You're right guys are jumping sitting here for Steve rose the gold report As a precious metal gold is still king It continues to hold the most effective safe haven and hedging properties across the global major trading hubs of the london otc market The us futures market and the shanghai gold exchange the gold report Tom o'brien publishes his weekly gold report every monday morning for subscribers Consisting of coverage of the xa u hui gdx the dollar bonds the south african rand As well as 25 different mining equities with specific buy sell recommendations The gold report New subscribers get a 30 day money back guarantee. 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That's tfnn.com then hit watch tiger tv So this is bowser chaplain sitting in for the hour of steve roge is away Catapult so the question came in catapult. Look at this All-time highs yesterday had a round number open of 331 And a nice green candle today. There's a round number open of 330 334 And it's moving to all-time highs. I don't know. You remember what we saw just now. Oh, let's go to arm arm Round number 164 all-time high. Where is it right now? It's at 142 144 was the round number high yesterday. There were so many stocks with round numbers that actually took them out Last night the closes of a whole bunch of stocks were at round numbers even though the official close was With a decimal They had in the left column right here where it said close They had round numbers. So something's going on and I I'm just going to say Um, I'm watching this very closely very closely. Next question came in Uh, did I finish fives? fis Oh, yeah, so that's what I said 65s should be supported this big green candle now What happened if this closes towards the high of the day that'll be marabosa candle no wicks or virtually no wicks That means For the next two bars. I look either if there's another a stronger gap up move. That's fantastic Normally you get either a doji candle or a pullback to give back some of the last hour's gains That's normally what I would see on a candle like that. Next thing we want to look at is postage pst But it's got nothing to do with postage. They took the stamp. Oh, it was stamp. It's dmp postage Spectac I followed this for so long pure storage ink A shares Unbelievable up 3.65 percent of that for dollar 62 54.57 Had a must have had great earnings yesterday spectacular move I have to call this huh I have to I I think I have to call this an a And this is a leg d in the month in the weekly chart leg f in the month And even that has to change because it's just so strong Um fabulous move. So all I could say is there are so many Moves to the upside that are off to the gap. That's usually really positive But two things you've got to keep in mind within the um within the context of Going sideways that's one thing but in this cut formation as a breakout What you do is that little the the lip on the left side right here down here below 45 to 41 That is major support Now you have to look for secondary support and the secondary support is the most obvious one Which is the gap low of yesterday where it opened it opened at 47 58 so it's It's at 54 right now. I put it here again I put it in the middle if it starts to close under 50.30. Let's say if it closes under 50 and I I'm I'll have to I did this quickly the other day because I really I looked at it as a potential because it's always on my I don't know if I've got it back on my list, but it's something I've always Noted that in packaging. They do two things. I would packaging is p k g I don't know if I've updated it. Yeah, look packaging. This is packaging No, no, no Oh, wait, wait. Oh p h k. This is p k g packaging At an high all-time high as we speak. So this is packaging core of america. I like I always The reason why I look at postage is because I always thought it was kind of This postage stamps right so I realized that the symbol that I'm looking at is different to the actual company and but packaging core p k g is Really important because it tells you about The proclivity of people to be buying things over and over and over. That's why amazon's up at the highs And this is a really it's in make sees a very bullish sign a Alternate count. This is a leg d in the weekly alternative count the last d pulled back quite a bit. Yeah, this is nice a b c d So packaging core of america is I put it together with With my cta cta s Wm waste management So this is this is america. This is what's going on in the economy right now So it's really there's a bullish side So let's go to my the question that we're looking at here. The next one was could I look at Microsoft so we're looking at microsoft earlier on And what I'd said was looking at those pd's. Let's see if I've still got microsoft. Yeah, so here's your pd In the 10-minute chart and to pull back but that nine is still holding well So for the nine for the green to turn pink you'd probably have to see this is a 10-minute chart Microsoft trading below 411.20 and that'll make a dreaded h pattern. Oh, I mentioned something that I'll have to show The dreaded h is right here dreaded h is It's when you come down sharply not you when the price comes down sharply And then makes a peak a ruby it fails and it takes out that left side low That's a that's a dreaded h And if it takes it out you can go one to one to the downside of the arch So I'm only watching this closely right now the nine's still holding very nicely. Okay on a shorter term basis Microsoft has gone right to the 200 period expansion moving out. So let me just show you this as you're starting Yep, this new start right we look for the lowest low to start your wave count peak a And every peak above it gets counted. That's a that's also an a that's now a b That becomes a c that becomes a d and that becomes an e and It pulls back pulls back and has one spike to the upside. That's an f That that didn't go pink. So I'll just put a pink Plus sign right there. I'm not going to do that now I'm a time and I put a down error and now we're watching this closely So the one minute chart is still in the down mode Um, and you're watching to see what happens with the 10 minute chart is still looking good. All right, so it gets out of this Let's go on you want to be looking at the next question came in cvna This is another one that Folks look at and they shake in and they say wow, I could have been buying this way down In the pennies and here it is at 80 dollars. So it's an old egg seat stopping right at a line is not being at since 2022 when it went negative Right there and look what happens. You can't strongly 200 period moving is as a 200 period moving average is as a magnet Back in 20 the end Let me see if it's december january Back right there February the week Week each idea week of february the fourth carvana cvna closes under the 200 period moving average for the first time since it broke above it back in uh, april the week of april the third 2020 it goes peak a b cd e a b c d e and then it has this kind of a b See but that that's see you have to put into context here Uh, let me just jump and take a b See Yeah, so when I get a c like that, I always go back and say what did I miss so that is a c It's probably the c and it's a minus fails very very seldom fails in the c So it comes back down, but then it went negative right there and the week of 21st of january. It hasn't been there since 2022 it's just touched it for the first time. That's how important the 200 period moving averages I'll be back basil chap is sitting for the hour of steam road a lot of questions coming in and even more. I'll be back Currencies commodities and bond markets are as important as ever right now with how they're driving the volatility in equity markets across the globe Which is why it's a great time to try out teddy keg stats tiger 4x report Teddy keg stat breaks down the 4x markets every monday using his 30 plus years of experience as a trading veteran of futures 4x stocks and options Teddy releases his weekly tiger 4x report every monday morning with coverage of all the major currency pairs Including the dollar index the euro dollar pound dollar dollar swiss Dollar yen as well as many more and he also has weekly coverage of the crude oil market and the 30 year t bonds as they both Influence 4x markets tremendously when you sign up for the tiger 4x report You also gain instant access to teddy 60 minute webinar archive He just hosted 4x strategies and fundamentals What is behind the tiger 4x report for all the details and to start your 30 day tiger 4x report subscription today Visit the front page of tfnn.com tfnn educating investors Everything in the universe is governed by the fibonacci sequence This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market To stay on top of stock patterns You can take advantage of sign up for the fibonacci 24 7 newsletter at tfnn.com When you subscribe you'll get a weekly report from veteran day trader lary pesavento on stocks You need to pay attention to and you can trust larry's analysis After all he's got 45 years experience as a day trader Larry will also provide daily charts videos and data on the key markets that he's tracking Expect notifications from larry on market movement. You need to act on at any time First time subscribers also get a 30 day money back guarantee If you're not satisfied let us know and you'll get a full refund within 30 days of signing up Subscribe to the fibonacci 24 7 newsletter today tfnn.com educating investors Will the smp 500 continue to climb for bold trades on us large cap stocks in either direction spxl spu u or spx s Directions daily smp 500 bull and bear leveraged etfs Direction leveraged etfs an investor should carefully consider a fund's investment objective risks charges and expenses before investing A fund's prospectus and summary prospectus contain this and other information about direction shares to obtain a fund's prospectus And summary prospectus call 866 476 7 5 2 3 or visit direction investments dot com A fund's prospectus and summary prospectus should be read carefully before investing An investment in the funds is subject to risk including the possible loss of principal The funds are designed to be utilized only by sophisticated investors such as traders and active investors distributor four-side fund services llc This program is brought to you by vista gold traded on the nyse american and tsx under the symbol vgz Our first basal champion here sitting for steve rhodes hour The dollars of 18 s visa 21 um We're looking at carvara. See our cvna trading up $4 at 79 94 made a high just the other day P e it's pulling back just a little bit now it's going green But the nine is look the price is way over the nine 90s way over the 14 period moving average in the daily chart MACD is still expanding the histogram. It's just now flattening out But it's the green line is expanding with the pink line the night the unbalance the Relative strength is still good. It's not great, but it's very good The stochastics at 82 percent that's great unbalance forms a little bit overboard But it hit the 200 period moving average after years That is a magnet line Because it's the first time you could hang around you and then it could pull back Towards the 65 area at some point. Well, it depends on how high it goes in this phase But it's a leg c and that's really good. That is the magnet line 79 right there 200 period moving average of 80 point 96 So it can go higher. It can go lower. It'll keep coming back to this area as I see it It's an leg d already in the Monthly chart, but you always get d's coming off a bottom Otherwise, how do you go to the top? So this is very bullish so far But very short term. I can see a storm to meander There may be 85 This even go to 86 as a high and I'd say 70 72 is the key support on the very short term watch it closely, but so far it's looking good But it isn't at the new high So it's taking three days to rest up at this particular point next question came in I did that I did that ARWR ARWR I'm looking at this. Oh, what a nice move today. This is uh leg b So this is a brand new leg b made a peak f top just at 40 or just under 40 It plummets down to the 28th 29 area now. It's peak a peak b The mag D's good the nine's good nine over the 14 the stochastic's weakest No, stochastic is an 80. I want to say the on balance volume is a little weak on the regular strength Is climbing but uh not as well No, I should I should say on the register the Registrate index is good and the weekly chart isn't good. So what am I looking at ARWR? Gosh, I used to have this all notated. I can arrowhead farmer Uh, yeah, this is good. I think it's heading towards 3782. That's the Georgia period moving average if it gets there next week I'm doing with speed now because that's what biotechs do if it gets there next week then the high that was made Of 39 83 the week of the 12th of january That becomes not just a target. I suspect it'll go to a leg b above that Key support is where it was yesterday at the close and that's right on the 9 and 14 period moving averages And that makes the 32 air 30 through 30 to maybe 32 30 to 20 that's kind of key support short term Then I had a question. Oh, I didn't really want to discuss this um Yes, so comics gold So oh comics gold. So what is going on? I have to suspect I suspect it's part of Middle East thing Because that's the way that those goals has been moving lately Um, it's not the dollar. I think it's now gold the fact that it's broken that left side support The the the gold stocks are so low That I think that you could nibble on some at this particular point I'm sure when tom does his show later on today at three if he's if he's on Um, they'll be talking about the gold stocks But this is the first time because of that 200 period moving average the break away with an inverse head and shoulders This is the first time that i'm actually seeing Strength and we've got to wait. It's not even 12 o'clock yet. So we really have to wait until later in the day But if I'm going to let's do this if on monday The weekly chart extends higher And tags the high that was made Back on the week of the 29th of december, which was at 2117 I mean theoretically that should be a long way to go but look how quickly we've moved up here Um, and it's only a leg b and I've got the stochastic for the first time at 80 Yes, 82. That's the first time I can say this is a move that could have legs and the On balance volume is working out well Regulus jet has been rising throughout this pullback and now with the rally And you've got the nine over the 14. Yeah, I like this very much even if there's a pullback I think that the trigger off the low that was made makes the 2026 to 2038 level Really good support if there's somehow other, you know, gold does that when you least expect this the first friday act and remember When we're looking at gold acting so well Let's just see Oh Nvidia so Nvidia Nvidia Very good move and I I'm not going to say anything other than it opened at a round number 800 all I'm saying to you is that I don't ever remember seeing the numbers that we're looking at of leaders That have round numbers in desperation where people are saying I don't care. I've got to have just grab it 800 take it it's mine and That just says to me that that is a form of round number hysteria And I think there's a penalty to be paid for it It doesn't have to be immediate, but it's saying to me. We're going to look back at some point I don't know about the market, but I'm saying that these astounding moves to the upside You will find that at some point we've given back a big chunk. Look, yes and video I've still got the letters remember I do every single charge the thousands of charts that you see a hand chart Doesn't take me long to do but it's a hand chart and then I lose a bunch because why I lose a bunch a bunch because um the When when it shuts down for instance yesterday out of the blue, I just had to shut down Comcast I don't want to do us Um and shut down but it came right back But normally in the old day and I've got all backups and anything But normally what happens if it shuts down and says you got to close it or it closes automatically I lose some data. It's there in the library somewhere But it doesn't come back like and I then I have to re-notate those charts Not everything but those charts that I lost So there it is. I'm pleased. I've still got the same chart. There's it was once I these it's astounding to me How great companies split And then they come right back to the previous price. Look at this that e means that the price was just below it Where are we right now? We're just below it in leg c At 5 819 in nvidia astounding move, okay So I'm watching it. I made an 823.94 top the mac d's good And mastic is good. So cast is only an 82 percent. That's good But not as good as it would be on balance forms a little bit overboard And do I call this a g just for the moment of putting in a g? I think I need to do some work because that was an instant restart That will make this e slash a f slash b and this will become a g I'm going to put a g slash c Because you never I never ruled out the instant restart Parallel people ask me how well if you got a parallel recap Wait, what are we supposed to what I'm saying is that once it makes an instant restart within three bars after d You've got no choice but to consider it in the chap and wave as an alternate count Uh, down's up 29. We've got the last break coming up. Gee, it did go quickly, didn't it? Did I miss any questions in the game? Yeah, okay, there's one. I'll I'll get to it in a moment I'll be back pencil trapper city in frisbee roads down's up to you You might think that if you want to be successful at trading in the stock market You're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life before you decide it's impossible get some advice from the experts You might find that it's not so impossible after all for daily market overviews that give you direction on the key indices Selective stocks and commodities subscribe to the opening call newsletter at tfnn.com The opening call newsletter is written by basal chapman creator of the trading methodology known as the chapman wave The chapman wave up down sequence gives you an edge in identifying price turns finding the peaks and valleys and stock prices Get the opening call newsletter by basal chapman in your inbox every day First time subscribers also get a 30-day money back guarantee If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up Educating investors Everything in the universe is governed by the Fibonacci sequence This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market To stay on top of stock patterns You can take advantage of sign up for the Fibonacci 24 7 newsletter at tfnn.com When you subscribe you'll get a weekly report from veteran day trader larry pesavento on stocks You need to pay attention to and you can trust larry's analysis after all He's got 45 years experience as a day trader larry will also provide daily charts videos and data on the key markets that he's tracking Expect notifications from larry on market movement You need to act on at any time first time subscribers also get a 30-day money back guarantee If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up Subscribe to the Fibonacci 24 7 newsletter today tfnn.com Educating investors Are you ready to take charge of your financial future? tfnn is your gateway to the world of trading and investing Whether you're starting out or scaling up tfnn empowers traders and investors of all skill levels with top-notch investing systems strategies and techniques It's time to protect and grow your money with insight you can trust Join us live monday through friday during market hours for exclusive content that moves with the markets at tfnn We bring the trading floor to you our season hosts are here to answer your calls and questions live on the air Check out the tiger's den for just one dollar and follow us on youtube and become part of our vibrant community And remember at tfnn. We're so confident in the value We provide that we are for a 30-day money back guarantee on all new premium newsletter subscriptions and services You have absolutely nothing to risk So why wait tune in live to tiger tv and transform your trading journey because when you know better you invest better Join us and experience the difference today tfnn educating investors RSP this is the s&p 500 equal weight etf. It means that they have to buy 500 of these little guys to get Whatever it is What it had you never see this in in etfs What it's an all-time high Uh, whoops 164 49 was the all-time high so within a dollar of an all-time high or so And it had 163 round number open. I mean give me a great count of that Apple apple is breaking down It's one of the few that's just fading to do what it has to do. So it's making the arch formation I suspect that you've got to have a laggard and every bunch of leaders is always a laggard. This is apple So that's part of what's dragging the down down a little bit. Um, we're also just real quickly. What was it? Apple Uh, yeah, oh, so yeah, so a quick question came in at tlt so that's bond So this is a good move up. It's m&a b I'm not yet sure that bonds are really going to scream to the upside just at the moment I think there's a move to the upside so yields can come down a little bit I just i'm not a hundred percent sure. This is not great action. It's really and I think gold is telling us about Uh, the gold is the um is the fear icon for international geopolitical events So that's probably what we're looking at here. Maybe that will affect the market. I don't know So in the meantime, I have a fantastic weekend Um, check out my opening call my daily newsletter. We grabbed a position today I had to do a split because if I bought it at the high it would have been ridiculous I needed a pullback it had gapped up huge And I had a split position because I didn't want to miss getting it But I also wanted to be buying it at the high and have it drop sharply because it looked positive So we got a split position. We only got one part of it so far. It's doing very nicely, but um Next week, maybe I'll get into it in my show a little bit about how if you're really missing something Start a position, but don't get the full position at least you're in it and you get a good feel for it So with that said, oh, this is the first time i'm actually waiting for the music. There we go Hey, have a wonderful weekend safely great program and still to come and check out opening call and my show at 10 o'clock Eastern time weekdays I'll see you on Monday


Good morning everyone. We've called this meeting because we have a very special guest joining us Who would like to recognize you for your amazing work? EMS week is just a week away and we can't think of a better time to recognize you or your hard work and Compassionate care for our patients in our community. So I'm excited to introduce number 98 and defensive end for the Tennessee Titans Jeffrey Simmons. Thank you for joining us today Jeffrey. Good morning. How you guys doing? We're good. We got a whole group here.  Well right now. I'm just um Taking I'm still taking my rehab.  I'm in need So like I said, it's always a positive to the negative size But like I said right now, I'm just working out and you know, trying to get my needle is stronger than what it was last year Awesome. Well, we're excited to see you play the fall Yeah, it's gonna be very exciting. You know, I'm excited to see how much I can improve improve and um, you know And help my team, you know, get to where we want to be and um, we got to just work together, you know Take game by game and and I think um, the everyone mindset Is it's right this year and we just you know, hopefully we just all work together and get to that super bowl games and win it Thank you enough for joining us.  You guys too. Tighten up Tighten up. Thank you


Good, that's part two, introduction to matrices. What we're going to do here is solve linear systems using gas Jordan elimination, Jordan elimination. So that's both exciting and ultimately boring. Now, for most of you, if you take a first course in linear algebra, it's probably going to start with solving linear systems or systems of linear equations to use the longer term. And boy, oh boy, is that boring stuff. Well, it's exciting in the first hand. But in the end, it's never something that you're going to do by hand. You're going to ask a computer to do it. And it's useful because to gain a deeper understanding of the mathematical properties of a matrix as a mathematical object, you need to be able to do gas Jordan elimination, but type the values into computer and spits it out and end of story. But yeah, you're probably going to have to pass a test on exam, and you've got to do these.  But yeah, exercise here is the key.  And they frustrate you because the tiniest little arithmetical error, which is very easy to make, and says nothing about your intelligence or your abilities. We all make little arithmetical errors. And then the whole thing goes pear-shaped. And then you have to do the whole thing again. It's just laborious and long. And I'm going to show you how to do it quick and easy using Julia. So if you're doing this for reals, you're going to just use the code. If you're practicing these for an exam or test or whatever the situation is, at least you've got Julia here or Python or the Wolfram language or whatever, just to check your work. And that's the beauty of what I'm trying to teach you here and of course. And our instance here, we're using what is a beautiful language called Julia. So in the Jupyter Network, as always, all these files, of course, on GitHub and links in the description down below, this is part two of this. I think it's going to be about nine parts just on matrices. Whereas the first part was just on vectors and I did that all, just in one go. Just this, what I've created here for matrices, that's just way too long for one video. So I'm breaking it up this time round. So to use Julia, we're going to extend its capabilities by loading some packages that excellent, wonderful, salt of the earth type people who sit down and use their time to develop open source packages. These are the saints and the heroes of the modern world. And we've got to thank them for their work and support them in any way that we can. And so we can use some of these packages. Two ones plots and the others, one's called row echelon form. But to do that, it lives inside of an environment that I created specifically for this course. And if I remember, I'll put a link in the description down below if you want to know how to set up an environment because every project that you work on in Julia just created a separate environment for it. Don't ever work in your base Julia installation. If that makes no sense, as I said, if I remember I'll put the link in the description down below or just search my videos, you'll find one about environments inside of Julia. And then also how to use Pluto notebooks. I hope I don't repeat all of this throughout these nine or so videos for this one. Anyway, I'm creating a computer variable there called file and I'm assigning that, remember equals is not equals as a mathematics, it's an assignment operator. And I'm assigning whatever's to the right of it to whatever's to its left.  And you call it something so that you can retrieve that information in your computer's RAM. And what I'm assigning to that is a string object and that string is a reference to a file on my internal drive here on this Mac. And it does see the whole address to that and what we're interested in is the project.toml file that contains together with a manifest.toml file information about the environment, this Julia environment. If it makes no sense, as I say, search my YouTube channel or check the links below. So from that we're gonna import the PKG, the Julia package manager. And in that package manager, we're gonna use the activate file. So we're gonna activate this environment. So we're just passing this file, the string to this file, to this functions and argument. And then inside of this environment, I've got at least these three packages, linear algebra, well that's built into Julia when you install it. Then plots which I've installed and then the row echelon, row echelon package, really neat little package to do a gas Jordan elimination for us. And you'll see gas Jordan, my pronunciation. Okay, and because we're using plots, plots is gonna create some very nice looking plots for us. And my personal preference is for the plotly back end search for plotly on the internet. It's a wonderful ecosystem for developing interactive plotting. That's the one, but plots, this plots function, the plots package, it supports a lot of back ends. The back end is what actually generates the plot here on your browser. So I'm gonna use the plotly back end. So I'm just calling plotly open close parentheses after importing the plots function at the plots library. So we can use the plotly function there by GR or there's a bunch of other back ends that it uses to. So solving linear systems or systems of linear equations. And this is what a system in equation one is a system of linear equations. So you see there are three equations and there's three unknowns. So instead of X, Y and Z, I've just used X sub one, X sub two, X sub three doesn't matter. Three equations, three unknowns. And the thing about a linear system, what makes it a system is there are values for X one, X two and X three that solve all three of those or how many ever you have simultaneously. So I'm not just looking for values for these three variables to solve one of them. The same set of variables must solve all three at one time. So here's an example of one of the solutions. We see X one can be three, X two can be two and X three can be two as well. So if I plug those in, I see it solves five, four, 10 on the right hand side. It solves all three of them in one go. So that's what we're looking for. And then by the by, I think I do it in this courses. If you don't have enough exercise material, just generate your own. So you have your coefficients, the one, the three, the negative two, the two, the negative two and the one and the two, two, zero. So do that, generate some random coefficients and then plug in a solution that you're looking for and then get the right hand side answers and then just hide them with the X one, X two and X three or whatever the case may be and just try to get back to these values three, two, two. See if you can find them again.  You don't need the books ones. But what are we talking about here? We've got these linear systems. We're all familiar with algebra and this is at school how you taught to write these linear equations. Okay, if there's only two equations, you are gonna write Y equals three plus X. But usually in linear algebra, we bring them all to the one side and leave the constant on the right hand side.  We bring them all up to the one side but that's algebra then equation one. That's what I'm trying to say. But what we're gonna do in linear algebra we're going to just rewrite those three equations in a very succinct way. And then a succinct way of writing it and working with it is to write it as a matrix. So just a little reminder because everything boils down to vectors. Remember what a vector is. Here we see the vector two, three. Right up there, the X value is two, the Y value is three and you can see this is a plotly generated one because it's all interactive when I hover over it or I can take things away and I can zoom in and I can go back home to see the whole thing. You'll see later on we're gonna use plotly. So this is not a course on how to generate plots. I do have videos out on the use of plotly for various languages because you can use it in Python other languages as well. This is how to generate that. That's not all that about it. Just wanted to remind you of what a vector is. So, you know, it's this values two comma three in this instance. So from zero comma zero, it's always a position of vector. So it starts at the tail of zero comma zero and goes to the point that you're doing because rewriting this equation one towards the linear algebra thing. If you think about what's happening here is just this equation three. I've taken out the coefficients. Look there, one, two, two, three, negative two, two, negative two, one, zero. And there they are, one, two, two, three, negative two, two, negative two, one, zero. And on the right hand side, five, four, 10. And I've written them in a different way. And the way that we've written them here is you'll notice that they're four column vectors here. And that's why we started with vectors because everything comes back to vectors. So you notice that they're these three column vectors. Now, I've just got a, we're going to do matrix algebra, how to do algebra with matrices. That's coming, but as a sneak peek, that's why I think I told you before that in the previous video that there's some things you need to know to work with this, but to need this, you need to know that. There's kind of chicken and egg thing going on here. So before we get to that proper, I just want to tell you, and actually it is a reminder, I think we spoke about it in the, definitely spoke about it in the video on vectors, the first video in the series. A scalar times a vector, just means you take that scalar and you multiply it by each one of the elements in the vector. And so in equation four, you see what matrix vector, scalar vector multiplications all about. And that's exactly what we have here. Scalar times a vector. And if I multiply this first vector, each element by x one, I have one times x sub one, two times x sub one, three times, two times x sub one. And the same there. So if I multiply all these out, I'm actually just left with that. So then when you see these for the first time, it's nothing special, nothing special is going on here. We're just rewriting things in a different language basically. We're just using the language of, instead of algebra, we're using the language of linear algebra. So, and now we see something very beautiful come out here. I mean, this is profound if you think about it. The system of linear equations, a linear system is a combination of vectors. That's profound.  I'm just summing up constant multiples of vectors to reach another vector. Yeah, it's quite interesting. That's profound. That's what the crux of this whole matter is. So I've got problem one here for you. Create a column vector and multiply it with a scalar, just in a little bit of exercise again. So what we've done here is create a computer variable called vector underscore one, and we are assigning to that whatever's on the right-hand side. And on the right-hand side is an array. I've just used a different way of creating an array here. I'm using the reshape function. And the reshape function takes in this instance three arguments. The first argument is an array of values, one comma two comma three, and I'm saying reshape this as something that has three rows and a single column. And summing with a single column, that's a column vector. So there's my column vector, three times one column vector of 64 bit integers. And in this instance, it said multiplied by any scalar. So I've multiplied it by three years. So the scalar three times the vector one, and that just each element is multiplied by three. So from one to three, we go to three, six, nine. One times three is three, two times three is six, three times three is nine. Simple as that, or this is something I want to show you. So in the last step of this language transformation or translation, I'm translating things from the language of algebra into the language of linear algebra, which basically means the notation, is this is the last way, the last step. Now we're not gonna work with this form just yet, it's coming. But what I have here, instead of writing it this way, which makes intuitive sense now, I'm hoping, is this sort of weird way. I have a matrix of only the coefficients, then a vector of the unknowns, and that equals the vector of the right-hand sides. And in matrix form, this is A, this is my matrix A, this bold. So if we look at the code here, the large tick code, it's backslash math BF, stands for math boldface B, and there's a math boldface X. So it doesn't print out well on the screen here in Pluto, but that X is actually in bold. So if you use Microsoft Word and you highlight something and you hit the B, little button to turn it bold, like this equation one is up here is bold and matrix of coefficients is bold. So what we have here is so that X and B must be bold, it just doesn't print out so nice and boldly here in large tick, but anyway, maybe they'll change it. AX equals B. So I have a matrix of coefficients, a vector of unknowns, and the vector of the right-hand side. And I can go even one step further and create this thing called an augmented matrix. Well, let's go in here and let me show you. So it says the augmented matrix. Let's make that bold by putting two underscores before and after. And if we run that cell, it'll take a while to run because they were all pre-rendered. Now augmented matrix is in bold. Okay, because I put matrix of coefficients there. So that's the matrix of coefficients A there, but this is the matrix, the augmented matrix. And what we've done here is we just dropped all superfluous writing. I don't write X sub ones, X sub twos anymore. I just have the matrix of coefficients on the left-hand side here, these three by three here and on the right here. So this becomes a three times four matrix and that's the augmented matrix where I include the coefficients and the right-hand side. And that's the one that we're gonna work with. Now that's a proper matrix and we're going to do something to this matrix to solve our linear system. We're gonna get back to those results. These ones up here were there three for X one, two for X two, two for X three. We're gonna get back to them by working on this augmented matrix. So you take your linear system, doesn't matter how many rows, how many equations, how many unknowns you write it as an augmented matrix. And I hope you can clearly see the coefficients and the right-hand side. Some of your lectures might even want you to draw, I hope you can see the pointer, the mouse pointer on the screen, a little line down here to separate these two from each other, because they really want you to understand that on the left-hand side is your coefficients and this last vector here, column vector here, is the right-hand side. But you needn't do that. That's, you know, as long as you're, I mean, of course you've got a brain between your ears, you understand what's going on here, but you know, if you're forced to do it, just do it. And what we're gonna do is we're going to do these three things. We're gonna, what we call elementary row operations. Now the column operations as well, and they work exactly the same, but the easiest way to understand this and the way that you're gonna be introduced to it, most probably is to do these row operations. Now what I wanna show you just, if you scroll right down to the bottom, so we're not gonna discuss this here, but I've put this little appendix here at the bottom. And what I'm trying to do with this appendix is to show you that what you did in algebra class in school was exactly elementary row operations. You just kept it in the notation of algebra. You kept it in algebra and you tried to isolate one of the unknowns so that you can back substitute. But what you did was really what we call elementary row operations. So all this stuff you did in algebra, the only thing that we're gonna do here is we're just writing it in a different way so it's shorter to write. And we're going to do these algebraic things just in a slightly different way because it's easier because we're writing things in a more succinct way. But all you're doing in this guys is the good old fashioned algebraic manipulation in solving more than one unknown and more than one equation. That's all you're doing. There's no difference. There's no magic going on here. There's nothing untoward going on here or some weird thing. That we're writing things in a different way. That's all. So read that appendix there. If you want some intuition or just this proper feeling of moving from having done it in school algebra form to doing it in the linear algebra form. So the three elementary row operations that we have, that's what we call them, elementary row operations are interchanging rows, multiplying a row by a constant and adding a constant multiple of one row to another. So that first one is very easy.   The order in which you write those three equations there that doesn't matter. Multiply a row by a constant. Now all you're doing is if you say multiply a row by a constant, all that is is you're just multiplying both sides of an equation by a constant. If you have an equation y equals x plus two and you multiply both sides by three it becomes three y equals three x plus six. What you do to the left-hand side, you do to the right-hand side and that's what it means. Multiply a row by a constant. So if you've written it in this very succinct form of equation six, this augmented matrix, if I take any one row here and I multiply throughout by three all you've done is you've taken that single equation and you multiply both sides by three. That equation which is now represented by a row, hence we call it elementary row operation and the row operation is multiply a row by a constant. Okay, and then the last one is a bit more, it's probably a bit more difficult. You're adding a constant multiple of a row to another and that I explain in the penics also I think quite thoroughly and think about it. What does an equation mean? Equation means the left-hand side equals the right-hand side.  If you added the three to the left-hand side you add three to the right-hand side. Okay, then the thing is still in equation. It's still equal on both sides. As long as you do to the one side what you do to the other side. Now if you have another equation of course the left-hand side looks slightly different from the right-hand side, but they're both equal. So I can add those two different things which are actually still the same thing because they're part of an equation, left-hand side, right-hand side. I can add it separately to two sides of another equation but because those two things are exactly the same they're like plus three on one side, plus three on the other side even though the side has x plus three and the other side has four on it, whatever they still equal to each other. So I can take the left-hand side of one and add it to the left-hand side of the other and the right-hand side of the one to the right-hand side of the other. I have not changed the equation at all because I'm adding to both sides something that is equal to each other anyway. So I hope that made a bit of sense. So add a constant multiple of one row to a constant multiple of another row. So what I'm saying is we're going to take one of the rows and we're going to multiply throughout by a constant so that means I'm doing to the left-hand side is what I'm doing to the right-hand side and then I'm going to add it to another row. So let's go through an example and it's going to make sense. Where are we going to end up? What we want ultimately, so I'm just going to show you what we ultimately want. We want this sort of thing. That is a matrix in what is called reduced row echelon form. So believe you me, we're going to go from this one using these three elementary row operations to this one. We have these, look at it, we have this sort of pyramid scheme going on here with ones going down this diagonal. Below every leading, and these are called leading ones or pivots, we're going to call them pivots. Below them is all zeros and above them is all zeros. But because remember these are coefficients. And what this, so let's take that first row for instance, that's going to read one times x1 plus zero times x2 plus zero times x sub three equals x sub one. So we're going to have actual values here and that means we can just read it off. That's going to be the three solutions here on the right-hand side. And what I've done is I've eliminated in school algebra, that's what you try to do. You try to eliminate one of the variables so that you have it isolated so that you can do back substitution. So this is the ultimate formula that we're aiming for. So here in equation eight now, this is going to be slightly difficult. I mean, it's probably best to write this on a board or by hand and record that.   I mean, as I say, it's boring and fun at the same time. One of those weird things to do these. And I'll make some of those videos. So this is purely for your reference. I'm going to go through it and I hope just it translates sort of what's going on the screen here and it being the static. This is really more for reading, but I'm going to go through it. Anyway, so let's look at this one. I'm going to jump back and forth. Look at this first one. So what we're trying to do, you see that leading one there? Remember, one of the things is we can just rewrite the three rows in any order we want. But one of the ones we've got a leading one there, so quite happy. I'm going to drop some tricks, tip some tricks here on you, the most important one that you can ever, okay, let me start here. There's not only one way to do this. There are many roads leading to Rome when it comes to elementary row operations. You can do this in so many ways.  You're going to land up at the same solution. So there are infinitely many ways to do this. One little tip is never, never do fractions right until the end, because one of your answers might be fraction. But then introduce fractions, because that just makes it 10 times more likely, well, that's just a thumbs up value, more likely to make an arithmetical error. So don't do that, just remember that tip, whenever you work with these, work with integers, never, ever, delve down while you're doing this to the level of rationals, fractions. Okay, so I see this leading one, and remember what I want, I want zeros below this. So I've got to get rid of that two, and I've got to get rid of that two. In the parlance of algebra, I'm trying to isolate for each one of these equations a single variable, so that I can just read off the answer on the right-hand side, because of the coefficients of the other two are zero, then they're out the game, you know? That's what we're trying to do, and that's why we're trying to get these two zeroes. So what I can do, one thing I can do is what I've done in equation eight. That first row, we've multiplied out by negative two. So let's look at it here. I hope you can see the mouse, otherwise this is gonna be weird. So if I multiply this by negative two, one times negative two is negative two. Three times negative two is negative six. Negative two times negative two is positive four. Five times negative two is negative 10, and that's what we have there. So I've multiplied that first row out by negative two. So in effect, remember, this is just short-hand notation for that original linear system, so I'm just a simpler way of writing this. So I've multiplied the left-hand side of the equation, these first three coefficients by negative two, and I've multiplied the right-hand side by negative two, and that's what I'm left with, negative two, negative six, four, and 10. So I've done an elementary row operation. I've multiplied one of my rows out by constant. Now in the next step, what I'm gonna do is I'm gonna add a constant multiple of the one row, which we've just done, a constant multiple, to one of the other rows. And that's exactly what we're doing here, because if I take row one, remember these first three values, that's the left-hand side, and as I said, that's why some lecturers want you to draw a line there to mind you of that. As I said, you've got a brain, you understand what's going on. So this is the left-hand side, and that's why I said these two things are exactly still the same. The negative two, negative six, four, equals the negative 10. That's an equation still, and it still has a left-hand side and right-hand side, and they're the same, it's an equation. So if I take these first three ones on the left-hand side and I add it to the left-hand side of another equation, and I take the right-hand side and add that to the right-hand side of another equation, that equation to which I'm doing these things, that's still legit, because I've done to the left-hand side what I do to the right-hand side. I've added equal, equal values to both sides, even though they're written differently. So if I do that, remember, the first coefficient is x1. So if I have negative 2x1 plus 2x1, I can add those two, remember this is algebra here. So I can say 3x plus 3x, I can add the x's because the 3x plus 3x equals 6x because you can add variables or unknowns that way. So they're neatly written in nice columns here. So negative 2 plus 2, now, before I go there. C row number one, I've just jumped back to the original form of it. So even though I've changed it here in equation two, so what I usually do in paper is I would write this negative 2, negative 6, 4, negative 10 on the side here and keep the 1, 3, negative 2, 5 in the original form. You can also think of between these two steps, equation eight and equation nine here, I've multiplied row one by negative of a half. If I multiply row one by negative of a half, I'm back to 1, 3, negative 2, 5 as it was originally. So I'm just doing two things in one go here because I'm lazy that way.  So aim for that, you'll naturally end there anyway.  So negative 2 plus 2, I can do that because they're both x sub one. That leaves me with a zero. The negative 6 and negative 2, that leaves me with a negative 8. So I'm just jumping between equation eight and nine here. The four and the one gives me five and the negative 10 plus four gives me negative six. So I've added the second row represents an equation with three on the left-hand side and one element on the right-hand side. And I've added to its left-hand side exactly the same that I'm doing to its right-hand side even though they're different things. But the equation one year, row one, they're still in equation. Both sides are the same thing. So it's all legit, legitimate. So now I'm also going to do that to row three in one step again because I'm lazy. So negative two and two is zero. Negative six and two is negative four. Four and zero is four. Negative 10 and 10 is zero. So look, I mean, we're getting there at least now under this leading one, I've got zero zeros. So in equation two, row two and row three here, in equation nine, I've just left with two unknowns. I've eliminated x1 from both of those. Okay, let's try and eliminate this negative four here. And as I say, that's why there's many ways to go about this. You can do these in any orders. What we're trying to always do though is work from the top to the bottom. Always work from the top to the bottom and eventually you'll see we get to the bottom and then we work our way back up. So what we're trying to do is to get these zeros underneath. So I'm very fortunate in this one and of course I designed it this way so it's easy. Is the zeros in one go underneath? So the next one will be this negative eight and it eventually must turn into a one but we're not gonna do that now otherwise we're gonna deal with fractions, keep these whole numbers. I've gotta make this negative four. I've gotta turn that into a zero. And one way I can very quickly see is if I make this an eight, I can add the second row to this third row and negative eight plus positive eight is gonna leave me with a zero. So to get this negative four into a positive eight, what do I do? I multiply by negative two. If I take this row three and I multiply by negative two, I'm left with zero, eight, negative eight, zero. And that's exactly what we have here. And now I'm going to add row two to row three. Oh, now I've got to tell you this. I like to do addition and many of your lectures are going to try to make you do subtraction. So they're gonna say subtract one row from another row. And I think that makes it needlessly difficult. I didn't like that at all. That's just, you know, just increases your likelihood of, as far as I'm concerned or for me, let's just be very mathematical about this. There's one example for whom it makes it more difficult. I like the addition, but just watch out. So I'll make you rather do subtraction. I like that. So if I take the second one and add it to the third now, I have zero and zero is zero. Negative eight and eight is zero. Five and negative eight is negative three and zero and negative six, negative six and zero is negative six. And there we go. I have under this leading one, I've got all zeros and this negative eight is not a one yet, but at least under that is a zero. And look at row number three there. Boy, oh boy, we've eliminated two unknowns. We've only got a single unknown left. And what we can obviously just do there is multiply that by negative one over three. If I take row three and I multiply by negative one over three, I'm left with this, zero, zero, negative three divided by negative three is one, negative three divided by negative three is two. And I can need a solution here, my first solution, because it says zero times it's except one plus zero times except two plus one times except three equals two. In other words, except three equals two. I've already got one solution. And remember how we designed this thing? I showed you one of the solutions before and two, what's one of the, is the solution. So if I eliminated things by getting towards this reduced row echelon form, and what we actually have here, it's just called row echelon form. So row echelon means I've got starting at this position one, one. Now I can go down a diagonal there and I'll have values there that needn't all be one, but below all of those pivots. So that one is a pivot, negative eight is a pivot, one is a pivot, below each of them, of course this last one doesn't have anything below it, but below the negative eight there's a zero, below the one there's a zero, zero. This is called row echelon form, and the process that we followed through the use of these elementary row operations is called gas elimination. It's called gas elimination to get it to this form. And now that you have a value, you can actually just back substitute because you know now what except three is you can jump back to the world of algebra, but that's not what we're about. We're gonna continue with the linear algebra thing. And as I said, we're gonna move back up and the way to move back up is to start getting zeros above these pivots. Now that first one doesn't have a equation above it, so that's not of concern, but we're going to work our way in reverse back up. So in reverse meaning this leading one here, I want to make that five is zero and that negative two is zero. So how can we make that five is zero? Well, we can take this row number three and we can multiply it by negative five. And then I'll get enough written that you have to use zero, zero, negative five, 10, which I'm now going to add to row number two. So there we go. That zero plus this zero is zero, negative eight plus zero is still negative eight. The five plus the negative five is zero and the negative six plus negative 10 is negative 16. So I've got a zero on top of it there. And I can also now quickly see that row two is looking very promising because I've eliminated both x one and x sub three. And if I multiply this by negative one over eight, negative an eighth, I get this, zero, one, zero, two. And that's very beautiful because now I know x sub two is equal to two. Zero times x sub one plus x sub two plus zero times x sub three equals two means x sub two equals two. That's a thing of beauty. And what we need to do now is to get rid of the other value on top of this last pivot, that negative two. And I think you know how to do that. We're going to take, again, row three, we're going to multiply it out by two, which means we get zero, zero, two, four. That is zero, zero, two, four. And we're going to add that to row number one. So one plus zero is one. Three times zero is three. Three plus zero is three. Negative two plus positive two is zero and five plus four is nine. So there we go. We have a zero above it now. But we've still got these two unknowns, x sub one and x sub two and row number one. So now we move to the next pivot up. We're working our way back up. And above it, we need a zero as well. And I think you know how to do that. We're going to take this row two and multiply it out by negative three to get zero, negative three, zero, negative six. And we add that to row one. So one plus zero is one. Negative three plus positive three is zero. Zero plus zero is zero. And negative six plus nine is three. And lo and behold, I told you what we're going to aim for. We have this reduced row echelon form. And on the right hand side here, this last column vector, and we're going to get to, it's actually just four column vectors, but that's a bit later. What we have there is our solution. X sub one is three. X sub two is two. X sub one is two. X sub three is two. And I'm just going to go in reverse because what we have now here is a constant multiple of a column vector plus a constant multiple of another column vector plus a constant multiple of a third column vector equals the solution. And that's what we have here. If we plugged in X sub one is three and X sub two is two and X sub three is two and we do that. We do the column, we do the scalar vector multiplication and we know how to add three vectors we are going to get to three, two, two. And there's something very deep going on here which we won't get to now but I'm pretty sure you're going to know what I'm talking about. Anyway, we'll get to it proper. So I have this constant multiple of these three vectors giving me the solution vector and that's exactly what we were after. Three, two, two is the solution that I showed you above.  Show you how easy it is to create them and just solve them. You've got to practice this over and over and what now comes the fun part in what I would call more in the modern world's reality not doing it pen and paper but just writing some code. So in the row echelon package.jl package there's a function called RREF reduced row echelon form is what it stands for and we just pass the matrix. Now I showed you how to create a matrix as an array. We pass it to the function RREF and it spits it out there. We see a reduced row echelon form and on the right hand side we see the solutions X1 is three X2 is two and X3 is two as well. So it's the real way we do it by using just a line of computer code but if you still have to pass your test and exams you just want to check your work. It's just as simple as that. Now though, just do a bunch of them get familiar with them and then I'll pass your test and your exam and then from there and it's all computer code. Not going to do this by hand again. I want to show you some interesting stuff though. We had a system, a linear system, a system of linear equations and three equations and three unknowns. That made things very easy. Let's add a fourth equation and see what happens because this is the source of all confusion here as if you have more unknowns and then equations or less unknowns and then equations. That's where all the problems come in. So now I'm just going to add another one X1 minus X2 plus two times X3 is five and that three to two is still a solution to this one. So it's still part of this linear system but now I've got four equations and three unknowns. If you do Gauss-Jordan elimination on that and instead of doing it with a pencil and paper or chalkboard and chalkboard or whatever, it takes your fancy as I say. If I do that, this is what I'm going to end up with. This last row is all zeros because it's redundant. It didn't add anything. I had three unknowns in a proper system and I'm just going to use the word in quotation marks here, the proper and the proper system. If I've got three equations, three unknowns, I only need three equations to solve the system and that fourth one is now redundant. It hasn't added anything. And if you do Gauss-Jordan elimination until a reduced row echelon form, you're going to get all zeros in that extra one, the one that doesn't add anything. So just remember that. So let's just delve a little bit deeper into this idea of more equations than unknown or the other way around. So what you, you know, let's start and for simplicity's sake, I'm going to start, I'm going to stick to two equations and two unknowns because that just makes my life easier. And again, as I say, I'm lazy that way. And I'm going to start with how you see it at school.  Remember, in any algebra, we bring all the equation, all the unknowns on one side, so that's going to come, become ax minus y equals negative b and as I write cx minus y equals minus d and we write the coefficients as an augmented matrix. So the a negative one, negative b, c negative one, negative d. So just remember that little equation because in your test and exam, you're probably going to get this as, you know, school algebra equations. You've got to sort of transform that into all the unknowns on one side and from that, you're going to create this augmented matrix. And so if you just keep this little equation 18 year in mind, you won't ever make a mistake. So let's do in problem four. So what we see in problem four here is two equations and two unknowns and I've already bought them all to one side, but you do realize that that first one is just y equals x because if I take the two x to the other side, it becomes negative two y equals negative two x and divide by or multiply by negative half, it becomes y equals x. So that's just a straight line y equals x and you see the second one there. If it is two equations there and two unknowns, remember an equation with two unknowns, that's a straight line here. We're dealing with linear and linear algebra, meaning that we're not talking about x squared. We're not talking about x times y or the sine of x and the log of y, you know, these are linear equations. So there we do, I've drawn the two of them here with some code and you can look at the code there. But if I have two equations and two unknowns and they both, you know, the two unknowns meaning it's straight lines. If the three equations we're talking about planes in three dimensional space, but as I said I'm gonna keep it easy, they intersect some way and where they intersect is the solution to the linear system because I've got two values, you know, a value for x and a value for y that solves both of those together. That's what it means. So I want you to have that intuitive understanding about these linear systems as well. It's lines or planes or hyperplanes if we go to higher dimensions that intersect some way and if they intersect at a point, that point which is actually a vector if you think about it because I can go from 0.0 to that solution point and there's a vector pointing there. That's also, again, everything boils down to vectors and we'll get to that as well. So if I have a linear system and it intersects like this, remembering that the physical representation of my linear system is just this graph, then I'm going to get, again, I'm gonna call this a proper one because I'm gonna get a single solution. And that's the solution for this linear system. So if I write that as an augmented matrix that negative two, the two x plus negative two y equals zero, there's my right here. I'm doing a row reduced echelon form of that. And I see that I end up with one, one, zero is above and below it. So that's reduced to our echelon form and I can read my solution on the right-hand side, one, one. So x equals one, y equals one, it's a solution to that. And the reason why we have this one specific solution is because this linear system represented two lines that have a single point of intersection. So let's look at problem number five here. I have a linear system again, x plus y equals two and two x plus two y equals four. And if you're very quick, you can see the second one is just a constant multiple of the first. So what they really are, if I plot them, is two lines that are coincident upon each other. And if they coincident upon each other, you can see both of them there. You can take the one away by clicking on it, then just the blue one is there. You can take the blue one away as well, there's nothing there. So add the blue one and then let's add the red one again. They're right on top of each other because in disguise that's the same equation they just written twice. But this linear system has these two equations and later on we'll see is they're not independent of each other. One is a constant multiple of these other. But what you can clearly see from the graph of this is we have infinitely many solutions. There are infinitely many x's and y's that I can plug into the system that will solve the system because the two lines are coincident. No matter what x I choose, there's going to be a y that solves the system. So in the end, there's going to be infinite many solutions. It's going to go off to positive and negative infinity on both axes. And if I do this, gas Jordan elimination, this is what you're going to end up with. Again, this last one being all zeros because it was redundant. And on the first one, we can read off the fact that this is a straight line because it says x plus y equals 2 or y equals negative x plus 2. And that means whatever x I plug in, I'm going to get a y indicating the fact that there's infinitely many solutions. And then here in problem 6 is the fact that I have two parallel lines. So this is a linear system where the lines never intersect. We're talking of cladian space here, nothing special. And there's no solution to this. They never intersect. So what would that look like if we do gas Jordan elimination using elementary row operations? What is that going to look like? Well, it becomes nonsensical because look at the reducer echelon form here using the RREF function. I shouldn't hide it there. RREF of this augmented matrix. You see that this second row says 0 times x sub 1. I've used x and y. So let's stick to that. 0 times x plus 0 times y gives me a non-zero number. And that's nonsense. I mean, if you add two things that you've multiplied by 0, they both be 0 and 0 plus 0 is 0. So the right-hand side can't be a non-zero value. That is nonsensical. And that tells you that this system has no solutions. There's no solutions to the system. It's an inconsistent linear system is what we call it.   And then, more importantly, and much more fun, come check your results just using RREF. It's as simple as that. But I want you to have this deeper into it. Start developing this deeper intuition of what's going on here. Many of you are going to start linear algebra by looking at solving linear systems. They're exciting. They're boring.  And enjoy linear algebra and see the usefulness of it and go out and use it because it's used everywhere. Linear algebra, for me, is perhaps more important than calculus. Anyway, that's a different story. So what we're doing here is starting to develop this intuition that it's not only good for solving linear systems, matrices are these mathematical objects in their own right. Just as a number is a real number, or a complex number, or an integer, or an irrational number, it's a mathematical object. And just like those numbers are, a matrix is also a mathematical object.  I mean, it's the greatest amount of fun you can have probably one of the in mathematics. So it's important to develop this love and this deep understanding of it so you can see what absolute power is hidden inside of a matrix. And we'll certainly try and get to all of those in this first two sections. The first one was on vectors, the second one now matrices, but I'm splitting the second one up, as I said, into multiple videos. So I hope you enjoyed that one. Leave a comment to ask me to make some other things, or these notebooks are going to be available to you on GitHub. Also, if I remember, put a link down below to show you how to install Julia and create an environment and use Pluto as we've done here. I've made videos on those. Many people have watched those. So hope you found this one helpful. Leave a comment. Please leave a comment. As important, subscribe if you haven't already done so. Leave a like. And as people make fun, proper YouTubers, not like me, but make fun. If you don't like it, hit the downward arrow's thumb or unlike the video twice.  Anyway, subscribe, leave a comment and ask me to do, if you want me to do something else. Let me know.


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