Okay this week the first one is U-Blocks. That's right. I love featuring U-Blocks. They have such good quality products. They've been making GPS modules for 20-ish years. They, you know, I remember when they first came out with their first series and I was like, wow, these are really, this is really cool that they have GPS units that are so sensitive. Their specialty is, you know, the high quality, sensitive, a lot of information. There's cheaper GPSs but they only give you NMEA data and U-Blocks will usually give you a lot more and they're a lot more configurable and customizable. So their 10th generation of GPS modules are out. The Max M10, there's the S-Series and the M-Series. Both are very tiny. I'll show them on the overhead later but these are, you know, this is like a centimeter by a centimeter. They're very, very small. They don't have the antenna built in but they're very powerful and like I said, this is the 10th generation. So they're just improving and improving on the technology. So the Max M10, like I said, this is their super miniature module. You do have to add an external antenna. However, you know, data does come out of it. You know, once you connect the antenna, it can connect to up to four different GNSS satellite systems and it's been compatible with some previous products. Not all of them. I mean, I know like some of them were bigger. They've gotten smaller since then compared to like the rectangular style that were popular for so long. So there's the M10M and there's the M10S on the right and they are slightly different. So the M10M, I mean, they're both basically the same core on the inside. They connect to the same GPS, GLONASS, Galileo and Beidou. So GLONASS is Russian, Galileo is European, Beidou is Chinese. I believe GLONASS is up. I think Galileo and Beidou are not completely up yet but I have to check because I'm not, I don't subscribe to GPS daily. So I'm not totally up on which GNSS systems are up and running. Both have UART and I2C. We'll take a look at that. The M10S has slightly higher sensitivity because it has a additional low noise amplifier and saw filter. It also uses a temperature compensated oscillator that's the T under oscillator whereas the M10M has the crystal oscillator. So it's a little cheaper but it's again not as sensitive.  Another nice thing is it goes up to 5 volts to 1.8 to 5 volts towards the M10S. It has a little bit of extra hardware in there, temperature compensated, only up to 3.3 volts but it gets you another, I think it'll show another 3 dB of sensitivity while tracking. So what it looks like and I think we'll look at the pinouts in a bit but basically inside is, you know, it's a microcontroller that handles the grabbing the data and doing the calculations and then converting it to the interfaces. So there is firmware running on it. The modules you can see here it has the RTC crystal and then the temperature compensated oscillator optional and then the low noise amplifier and the saw filter also optional on the S series not available on the M. There's power, there's VDDIO, there's VDDRF, there's a couple of power supplies and of course there's a V backup which is the coin cell or I think you can also use like small rechargeable alkaline or lithium coin cell, you know, solderable coin cells for battery backup to keep the RAM backed up so you don't have to re-download the entire Almanac each time. This is where you can see all this, you know, the specifications but basically the detail you want to look at, you know, for the difference between the two is in acquisition underneath sensitivity. The M10M has a negative 164 dBM and the M10S is a negative 167. It also has a slightly faster aided start otherwise it's pretty much the same. So just tracking is a little different otherwise they all have similar specs. And this is the pin out, very simple. I like how it's very straightforward. You can see there's a right hand side, there's time pulse out, that's pulse per second. There is the battery supply, the reset pin. Left has I squared C, right has UART, there's safe boot end because I think you can upload the firmware if you want, if you have custom firmwares because again there's a mic controller inside. And then there's an RF and LNA area which you'll talk about when you have an active antenna, you can use it to power the antenna and also do some detection. Okay so your interface, so you know standard your NMEA output is classic for GPS units. If you need that, you got it. This one is going to be default 9600 BOD but you can see it goes up to almost one megabit or down to 4,800. You know all these standard BOD rates that you can use of course if you haven't you know giving you 10 hertz updates and you have all the sentences enabled you'll want something fast like 115 or 230 kilobit per second 8 and 1. And then what I really like is they also have I squared C. I do like I squared C because you know for things like a Raspberry Pi computer or so even some microcontrollers they don't have an extra UART or it's annoying because you know UARTs are you have to buffer the data yourself so as each byte comes in you have to quickly put it in and into your buffer queue and then keep track of it whereas I squared C has its own internal buffer for I squared C and then if you see at the bottom there the register layout most registers aren't used you use registers fd and fe to read the number bytes in the queue and then you can just read continuously register OXFF to get data bytes in I don't know exactly how big the queue is probably like 64 128 bytes or something very handy you can run it at 400 kilobytes one thing I do want to mention I'm sorry kilobits per second there is a clock stretching interface so you know some some chips do not like clock stretching just something to be aware of you know earlier raspberry pies didn't do a great job with clock stretching you'd have to lower the frequency rate so you'll have to figure that out I don't know how long the clock stretches for but it is in my controller inside it's not a hardware I squared C interface these things are so small because they do not have a antenna like some modules that you might be familiar with and so you'll have to add either passive antennas what a passive antenna looks like in this case it has a ufl connector or an active antenna active antennas are powered by clean 3.3 volts biased into the rf line but of course you're going to get much better performance we know with the addition of the draws an extra like 10 to 50 milliamps whatever it takes this is an external antenna so which do you use it's totally up to you you get to decide because you get to put the circuitry in so this is what it would look like for an active antenna you see there is the antenna supply you have an RC filter and inductor to feed it into the antenna RF in if you're using a passive antenna you just leave all that out so super easy you decide which one it is there's also a antenna supervisor circuit which I kind of think is interesting and this is really handy because a lot of times people I've noticed people using you know they're using antenna and it gets disconnected or the ufl or there's a short and like my gps doesn't work it's very hard to debug a gps because basically either you get a fix or you don't so having an antenna supervisor is really handy it is a bunch of extra circuitry you have to add they do give you the layout so like this is the three pin circuit so it lets you detect open closed like shorted whatever is connected it basically does everything but it it does require you to see um you know comparator and a bunch of transistors that said if you want to build a really rigorous gps um into a product and you want to alert people that the antenna is not working or it's they maybe they connected the wrong kind of antenna um and the kind of thing is what makes your product a little bit better than just like hey you know either it works or it doesn't because again gps is so opaque like either you're getting nma sentences and either there's a fix or there isn't usually doesn't tell you why you're not able to get a fix um another thing that I thought was interesting as I was researching this is that they actually came with a kind of a cool idea I'm hoping I'm describing this correctly but they have things good stream things scream things stream which is their iot asset management you know servers because I do wi-fi and they do rtk and they do gps so it makes sense that they have their own iot back end that you can stream data and I believe they use mqtt but what's another interesting um thing is that you know normally when you have a gnss receiver right it's it's you're not getting the data from the satellites that's like telling you what your location is you get these you know time pulse signals and you take the calculation of the deltas and you figure out like okay based on like the movement through space and the almanac of where all the satellites are it'll calculate for you you can calculate your location by using triangulation however that does take energy and time and you have to have that almanac to know where every satellite is and so that's why it can take like 45 seconds to do you know cold start and then this idea which is like well what if you don't do that what if you don't have to keep track of the almanac you don't have to calculate you just as long as you can get a couple signals from three satellites um you can upload it to their service and their service will do all the math for you um so it's really really fast because you don't have to do a cold start and then as long as you do have some wi-fi connectivity maybe you have like a nbiot you know you can quickly um turn it on send some packets out of laura send out that timing data to their service and then it will calculate the location for you so it's an interesting idea because on one hand of course you have to have wi-fi or cellular connectivity but you don't have to keep track of as much like the gps doesn't have to do as much work and that's kind of an interesting idea like offloading the gnss calculation and then finally uh good timing there's actually a webinar that they're doing with digike in a week uh it's uh a week from now so sign up uh if you go to ublox or digike's twitter account they've just posted it it's also if you just google for ublox accelerate your wireless solution development um uses their explorer iot uh humidity and temperature sensors iam use sensorion uh micro e looks like the scd 40 or 41 there um it looks like a really cool way to get started with you know their their iot platform module that has wi-fi nbiot cellular laura bluetooth and uh it's got stem and qt and quick connectors on it so you can add more sensors and i think the webinar is free and you can pick up the explore iot kit which has one of these um i think max 10 gnss modules inside of it so it could be a really great way to get your asset track or gnss project up and running available at digike it's in stock that's right nine thousand of them it's over nine thousand um so they have the s in stock i think the m uh isn't but i will say one of the things is that like this wasn't this was not in stock when i first started the iam npi but it was like they were so cool i thought i would do it and then like two days ago suddenly they came into stock so i did get some i can show them on the overhead you want to show that video after they are yes but i just want to show this really quickly they have a cute little video but i just want to show how unbelievably small they are you don't have to let it refocus because it's it's so small it's so tiny um very tiny little module so they i mean it's amazing how small it is of course you'll need the antenna but the antenna usually goes on the outside anyways and it comes in uh tape and reel okay here's the video see on the other side yeah


Create your safety buffer. Simple rules like four months of expenditure. But make sure you've got something to fall back on that buys you the ticket to take risk. The rest, get it going. Put it into the markets. And the reason why people are so behaviorally or emotionally anti this, some of it is just procrastination. I just don't get around to it or I'm not that interested. But for a lot of people, it's fear. It's fear that if I put my money into the markets and the market's turned down tomorrow, I've failed and I'll just keep it in the bank account. And you hear these positions where all the stock market's just like a casino. In a sense, that's true. In the short term, the stock market is a bit like a casino. We don't know where it's going tomorrow or next month, et cetera. But here's the thing. If you go into a casino every day and play the roulette table day after day after day for years, you're going to lose, right?  This is the end.


Hello and welcome back to another trip with me to Psychoville, where Vice is the so-called premiere news outlet. And from page stories are insanely racist lists of 100 things white people can do to make life less frustrating for people of color. Unreal, no wonder they file for bankruptcy and are barely operating. Which is probably why they just reposted this rage bait from way back in 2018. So here we have Casina Boom, whose lived experience and expertise includes quote, womanhood, blackness, mixed race identity and LGBT issues with a specialization in lesbian identity. All that hard-earned expertise, yet still no self-awareness. Because this article is just one big batshit crazy projection. Now we're not going to suffer through all 100, but I will go through a few right after I tell you about this opportunity to secure your retirement and get a free coin while you're at it. Investing in precious metals may sound confusing, but the team at Noble Gold Investments make it easy. Let's hear from actual Noble Gold Investments customers. The staff answered all my questions and helped me every step of the way. No pressure sales tactics, just honest guidance. Securing my future is less stressful thanks to Noble Gold's expertise. So don't settle for financial uncertainty. They'll suggest options and see if you can diversify into gold and silver. Right now Noble Gold Investments is offering a free five-ounce silver America the beautiful bullion coin for qualified accounts. Don't settle for financial uncertainty. Noble Gold Investments has an A plus rating with the Better Business Bureau and countless five-star reviews. So why wait? Let's get started right now. Go to noblegoldinvestments.com now. That's noblegoldinvestments.com. The only gold company I trust. And remember, there's always risk of investment and there are no guarantees of any kind. Alright, as promised, the list of things white people can do to make life less frustrating for the rest of humanity. Because I guess we're just so annoying. So this article starts out.  How about new? Seriously, what happened to live and let live? Now white people have to literally strive to make the world less terrifying for the apparent superior people of color? That seems normal. Number one, just because you can't see racism all around you doesn't mean it's not happening. Trust people of color's assessment of a situation. How about new? Right, right. Just because you can't see the boogeyman doesn't mean he's not there. Just blindly trust us because of our skin color. We are superior and know better. Don't assume that all people of color share the same views. We are not a monolith. I know that these people don't seem to have any self-awareness about themselves or their actions.  Yeah, don't assume that a whole group of people share the same views and thoughts. That would be so bad. Number three, don't assume or guess people's races. This is not a fun game for us.  Now, is this something that white people and only white people do? Even if it's true, what's the big deal? A white person who would say something like that would be clumsily trying to be your friend, not terrify you. I'm starting to think that maybe the lesson here is that these people should stop trying to be your friend. Number four, if someone tells you they're from Uganda, don't say quote, I went to Nigeria once. Just please. That sounds like something very specific and weird that might have happened to you once and that you're now for some reason ascribing to all white people almost as if it's in our DNA. It seems to me that there was a word for that at one time, but they redefined it. So never mind. Strap yourselves in because these just get better and better. Number five, related. Don't refer to Africa as a country. It's a continent and it's widely varied. Yes, take a moment.  I know Africa is a continent. Lots of white people probably do. Maybe some don't. There's probably lots of black people who don't know that. That one is definitely gotta be what? Because it's like Idaho and all that, Tennessee and all that. Yeah. Right. So moving on. Number six, oh, and rest assured that literally no person of color ever wants you to get back from holiday, show off your tan and excitedly exclaim, look, I'm almost as dark as you. I went to that tanning place. Your wife suggested was that place the sun. This is just a deranged list of stereotypes and things that either happened to her or she imagined. And she's now attributing to all whites. Don't assume that a person of color knows everything about their country or heritage. Do you know everything there is to know about America, Germany, Sweden? That's what I thought. Wait a second.  Number eight, don't assume we could run if we're black. Do math if we're Asian. Have drinking problems if we're indigenous. This article is literally a list of assumptions you made about white people. Oh, that was different. Again, we see this proclivity of left-wingers to not hold themselves to the same standards that they hold their opposition, which this lady clearly sees white people as. Number nine, regard us as automatons, unique individuals, not as representatives of our race. Number 10, don't make embarrassing jokes to try and be down with the people of color. We'll laugh at you, not with you. Number 11, don't rinse our culturally specific memes. They're ours. Go enjoy that weird one about the plums. Look, I have no problems with any group of people, have had friends of all colors, and will continue to treat all people like I want to be treated. But damn, if Scott Adams wasn't on to something. All right, that's all I have for that. Thanks for watching. I hope you enjoyed and I'll catch you on the next one.


onder if burlington noticed too so um given the uh significant facility improvements of the school um there's no mention in here of solar or any backup energy energy uh resulting from that solar is that is that a capability result that energy sold to the grid i don't know the answer that um my guess though is that it's it's a small fraction of what they need so it would wouldn't like it might add to their resilience but i don't know yeah because i don't recall there being any battery backup um and given this recent holiday storm um and the power being cut uh yeah it just raised the question to me if that i think i got a deal because it all goes back in the grid that's kind of what i was from recalling as well yeah so elaine would you like us to adopt this resolution and then vote on changes for next year uh that would be great okay are there any questions from public public attendees or comments may or if i could i'm not public per se but excellent yeah i just wanted to um give kudos to elaine um this process was um awarded a contract from the state to an out-of-state contractor who did this update it's a new platform that emergency management push push the county into and this is a big big lift um it was through covid so that contractor was not able to come it was all trying to be trying to do it um over zoom um it was very clunky we had an intern manager at the time um so there's a lot going on for us and for elaine to come come here and then pick the pieces of this and i do mean pieces of this up and pull it together to what you see today was a huge lift so i just want to commend her appreciation for doing such thank you john can i don't know without you and john rousher and eric thank you um can i have a motion to approve the resolution as provided so second motion by jim second by aurora all those in favor please say aye all right motion carries and thank you for the comments we'll include them in next year they're useful yeah so do you want me to vote do you want me to run a vote on this um i wouldn't worry about it because you'll see them next time i feel like they're all reasonable so we're happy to put them in the draft they ask that we still do a vote on these given that i may not be able to vote on them next time and can i just add um i have a question to clarify clarifying question um page 28 on the critical facilities the historical markers and historic districts um we don't have any historic markers indicated in winewski and i believe we had recently approved one yeah that actually hasn't been installed yet it's been approved at least we're talking about next year right well fair enough yeah that's a good point let's make sure to remember that part of the and actually that map we don't think that that that um outline is correct so that's another one we'd have to revisit in any case yeah so all right i mean even just reviewing and updating page 28 indeed might be a better way to go about it yeah well it's a good point about the marker right you know it's good to call it those specifics that you're aware of okay um so looking for a motion to for next year update emerald ashmore f35 crash historic section to bring it up to current uh correct the map on page 26 of the city hall and fire correct the historic map on page 28 and note the school district uh renovation with solar um also which church and as a shelter actually don't and then the the history okay got motion by jim yes second second by aurora all those in favor please say aye motion curious thank you thank you all for the attention to detail indeed in a lengthy document yes item e public art spending okay this feels like a little uh luxurious but um that's timely in a way so mostly reading from my cover sheet and arts district is proposed in the city's master plan and economic development plan while public art is not a priority of councils for this fiscal year i'm asking you for direction now because of the potential opportunity this spring anecdotally there was an increase in graffiti in the city during the pandemic credit to councilor oak leaf to point that out to me very early on in my tenure uh studies show that murals can deter tagging and other so-called nuisance graffiti at least one organization is interested in doing this in wanouski the spring which is the opportunity referring to along the riverwalk but other individuals or organizations may also be interested and i would recommend any such work be paid for um so given that and where does that money come from um on august 1st 2016 city council established the kathryn deek dekaro wanouski public arts fund after uh two city managers ago to expand the presence of the arts in the city uses of the fund shall include um reading from um the enabling um art related opportunities from for residents to enjoy installations and exhibits to enhance the cityscape and the use of the arts to promote a sense of pride cultural expression and celebration of wanouski thanks to angela for digging that out and we have no existing procedure or policy to replenish this fund the current balance in the public art fund is five thousand five hundred and one dollars made up of some council pledges and a donation so i have two questions for you tonight if you choose to take them up um there's plenty going on so we'll be understandable if you don't feel like it um a there whether you would like to advance the public art initiative at this time using the current balance and if so whether you would like an existing commission or commissions to advise you on how to spend it thanks lane um this fund was established in 2016 it's never been used right that's far as i know angela um if there's like a community effort that's driving this in low lift to staff i'm not opposed to using this money and then having one of our commissions participate in that yeah i i think it sounds like a great opportunity i think with especially looking at the language of the pride cultural expression and celebration if this might be good for either inclusion and belonging or possibly maybe safe healthly connected people trying not to overload inclusion and belongings plate um or just popped in my mind i wonder even if the library committee might have ideas if maybe they they know folks in the community who might be interested in doing work like this um so those are those are just kind of the ideas that i come i came to mind and i don't know if council wants to dictate that or ask if any commissions would be interested in it and i'm not sorry go ahead okay um actually so unordered district that was proposed as an economic vitality measure so it could make sense there it also fit with me some infrastructure because some of it at least will presumably will be on the municipal infrastructure i was just gonna say i don't there's already somebody proposing something a community effort so i don't think we're like asking a commission to look for ways to use it for this particular instance but to be involved in or to provide advice on on this thing that already is out there yeah so my my impression is supporting the question is do you want to support this opportunity this spring as question one and question two is we don't have a procedure or policy at all do we want to assign something to one of one of the commissions um if any of the commission so that's that's how i'm interpreting what your question is at lane is that is that correct well so a nuance on question one is that there is an organization that's interested and they've a couple months ago they quoted me three thousand dollars that would almost deplete the fund right it would be more than half so part of my question is if you want if you wanted to proceed with that then i'm not sure how much more like christine said i don't know that you need a lot more input from commission and it would be a low lift and it goes on from there right so i'm seeing part of that is do you really want to would you want there to be more of a process rather than spend so much of one fund a one time fund on an organization without even a process of selecting who it is so i i would say that those are excellent follow-up questions um it sounds like this the intent was for this public arts fund to be something that continues well into the future and it and is not a one time but you know i'll say if there's additional history there that i'm missing you know please correct me um so i would almost see a the role of a commission is like how could funding be like two parts how would we want to select assign locations for our installations as well as like how to allocate or propose funding allocations for it um so i i guess i see that second part being in my opinion separate from the first ask i don't see this the second part reviewing the first ask does that make sense i call them okay so the question question one being do we want to do art this spring we already have somebody who has proposed to do a project and has already provided a budget estimate do we want a commission to review that particular project i don't feel strongly about that do we want a commission to review funding allocations moving forward and project like procedures guidelines moving forward sure yes i i do support that side of things i feel i feel the opposite i think we've got this money that's been sitting here forever unused we have someone ready to use it let them use it have a have a commission though have some input on like review their proposal i don't think it's a priority to develop guidelines or funding mechanisms for this fund given all the other things we have happening yeah i disagree well given that we have no plan to re-up it um any policies and procedures we would develop would be solid for the distribution of five thousand dollars which is a lot of some cost for a small pot i would be tempted to put this to the safe of my inclination is put this is a healthy connected people commission ask them to review the project and also part of that review question is is this the best use of this fund at this time or do you want to see a competitive proposal process like i think that could be a part of the review is like is this proposal in itself worth it and appropriate and well thought out and is it so well done that we should just approve it as is or do you think this is like yeah it's good but um it'd be nice to have this opened up to other groups in the community have the chance to pitch in for these funds i think they could be both of those at the same time and make a recommendation back to council because what i hate to do is walk away from the slam dunk and what i'd hate to do is uh just do the first and it comes across because we could so i think that would be a middle ground to um not doing either of those i think uh that would be my preference as well and my sense is that safe healthy connected people does have the would have the capacity to do that work at this moment maybe not this month but definitely next month i agree um two point i don't feel like it's makes sense to create procedures if we're not going to fund the fund um i i would finally advocate that we that we do consider for economic economic vitality purposes for municipal infrastructure purposes safe healthy connected purposes i mean it's it's um those are interconnected value opportunities um and i think it's uh it's something that could be considered um as you know that's something the commission can consider funding opportunities not necessarily that are exclusive to city allocation i'm sorry i'm missing the kind of what would they consider an addition or instead of um so there is not a cap that i know of with the um downtown wanouski allocation i believe that we just approved a five percent and so having a potential cap on um the the city's contributions in downtown wanouski could allow for a percentage of economic vitality funding to be allocated towards the public arts fund but is that the discussion that we need to have right now no it's saying it's something the commission can consider moving forward share it like not right now i i don't think we need to ask them that at all like i i don't think i'm not supportive of trying to fund this in an ongoing way right now like this is a discussion to to parking lot for some future date yeah i don't know right yeah i see it as if it was something to go forward we then start leaning on staff because i could see it being something that would be great to like get donations or grants for but that's a lot of staff work um so like like jim said i think we should take advantage of this opportunity do have a little bit of review process just to make sure you know there isn't i don't know something that comes that is flagged by the state healthly connected people um commission just to have that um second look and kind of float that idea of you know get it yeah if they think we should do a slightly different process for specifically the spring slash summer and then i don't know for some somehow in the existence of this fund for to come up when there might be staff capacity slash funding capacity in the future does that make sense like i love the idea of having procedure but uh also might be ideas taking advantage of an opportunity jim do you want to motion what you described um i move that we refer the proposal to the state healthy connected people commission and ask them to make a recommendation on whether this is a good current use of the funding whether they recommend this back to us for funding i missed the latter part you said whether i would move that we refer this proposal to the state healthy connected people commission for their review and as part of the review ask them to consider not only if this proposal is worth funding but if this is the right time to use these funds for this purpose that makes sense i would like to second i would like to second that motion motion by jim elaine the clarifying yes um this by this proposal do you mean the the the cost and project proposal that i've already received yes that's what i yeah yeah yes yes yeah so the proposal reference in the cover sheet okay motion by jim second by brinn all those in favor please say i hi motion carries thank you thank you on to item f equity update from staff okay so so again mostly reading from my cover sheet the budget recommended by the city manager that's me for the next fiscal year july 1 2023 to june 30 2024 includes pausing the equity director position to help get down to a 5 percent tax rate increase members of the public have asked what this means for equity efforts by the city the most obvious impact of pausing the equity director is that we will not have access to someone on staff with equity expertise and who is intimately familiar with the city in our efforts to create a more inclusive workplace and city however pausing the equity director does not mean all equity efforts would stop the city's first equity director resigned effective july 2022 because during the year she was here she concluded the role was not sustainable and experienced a workplace that was often felt unsafe or unwelcoming equity efforts have been underway before during and since her time with the city many efforts of significance like the equity audit accessible communications and the inclusion and belonging commission were made possible or enhanced by her the need to incorporate equity into all city departments was certainly made clear by her and staff have more work to do on that but many other efforts were initiated and are continued by other leadership team members in every department so i'd like to review those tonight for the sake of transparencies certainly some of these things i don't feel are that obvious to to members of the public so welcome the the opportunity to to share them the activities fit into the categories of accessibility voice miscellaneous welcoming and awareness and these are listed in order of number of initiatives or efforts per category so for example accessibility has the highest number of of efforts so i'll review a few examples a non exist a non exhaustive list some examples from each category that are recent under accessibility improving or adding sidewalks is one effort that has an equity impact we've coordinated with when you ski school district on sidewalk plowing prioritization for underserved households this was a direct result of a recommendation by the former equity director removing unnecessary qualifications from position descriptions this is a common best practice to improve access to to good work good paying work as i would or good benefit work as i'd say local government work is improved access to recreation program registration by reserving half of the seats for paper registrations and then in person support for that type of registration improving readability of public communications by using a reading level review aid continuing to offer hybrid meetings and then even just today the finance director angela d r i the hr director jesse acre and i met with a member of our extended muski community they had commented during my budget presentation regarding the proposed staff pauses with equity impacts and had reached out to me to offer ideas i do plan to revise my budget to incorporate their primary recommendation which is to bring in more representation from the community for city staff to hear and learn from and to compensate those community members for their time this is a model that has worked for that particular individual in their work on equity after hearing the public comments after my budget presentation angela and i had combed through my recommended budget again and did decide we could comfortably add $50,000 of local option tax revenue based on updated projections this year and past experience so $2,500 of this does go to downtown muski by contract as brun had just alluded that would leave $47,500 we can add to the pro the pro i'm sorry the professional services budget in the administration area for the purpose of developing this i'd call it a mediated community voice program this is really about i'm putting it in the accessibility bucket because even though you know it's a voice it just said voice it's really about increasing access to city government by historically marginalized people and i'll also underline that even the fact that this person was willing to spend their time meeting with us is itself an outcome of increased access i had previously met with them in my effort to meet with everyone really hoping for that everyone with a stake in muski which gave them some assure and said i would want to hear their ideas so again i want to underline and commend council's efforts to to get out into the community and and hear from the community directly that sort of effort is important and i think is yielding fruit the second category is of equity efforts i call voice so um those included that we conduct a listing sessions on how to spend one-time money meaning arpa with historically marginalized groups including new americans which were interpreted uh youth and seniors um that was a big effort um led by the equity director and of course the communications director paul starne we considered how to best engage people traditionally left out of planning processes and developing city plans such as in the bicycle and pedestrian plan that's upcoming well we've created and supported include the inclusion and belonging commission to create their own structure and procedures and this body was an outcome of the equity summit of 2019 so that even preceded um as jim reminded me the equity audit or the equity director we are piloting stipends for commissioners include um specifically with the inclusion and belonging commission to support more diverse participation on commissions to respond to a question from brinn unfortunately i had overlooked that council had not approved these stipends before i had to start paying them i was going off a grant agreement of $50 per meeting um i've asked the commission the inclusion and belonging commission to provide some input to council about what this means the per meeting stipend and if there's a better way to spend that and then we would be coming back to you hopefully with those recommendations to formally approve the stipend policy for commissioners um angela i believe we're currently paying them quarterly we are just for um making it worth issuing a paycheck for the the per check fee that we're charged um we are also not seeing every member accept payment um we do have at least one member of the commission that is opting not to take this stipend um and we have only paid out i believe less than $200 to date for the first quarterly payment thank you um so and then so that's that item so continuing in the voice item um or the voice category rather so planning with the equity consultant and the we are planning with um the winieski school district next steps in engaging the public on equity audit findings under the miscellaneous category the police department uh partners with howard centers community outreach team to connect people to social supports were involved in an interaction with the police department we've been reworking the pay plan to value um so-called caregiver positions which have been disproportionately filled by women and people of color and this was in response to the equity audit we've also um looked at allowing experience in lieu of formal education again in best practice and increasing access to good jobs um working with housing commission to propose changes to ordinances to improve quality of rental housing obviously that's been an ongoing effort um we or and I are participating on the city's behalf in ideal vermont which is led by the state office of racial equity it's a municipal learning cohort and participation of city staff as well as council although there's another gen item tonight for that um in the conversation about a school resource officer um that was in response to the winieski student man's for action in the welcoming category the the last two categories are much shorter approving readability of the personnel policy um so that once people are hired in they have a they can navigate that document more easily that's something the hr director is working on um placing the equal o employment opportunity statement at the top of job postings not at the bottom as has been common um I think that was actually a response to the equity audit and then um again in terms of welcoming it's a small gesture but some staff are listing pronouns on our email signatures in the awareness category we are conducting at the leadership team level meaning like the department heads um peer learning on equity topics some of the um other departments are also doing this and then equity workshops for leadership team members that were designed and delivered by the equity director along with partners so again just to close this um diatribe I guess the purpose of this in-gen item was to communicate to the public the types of equity work that are underway and that will continue to that will continue an illustration of what will continue it's not the fullest we do welcome feedback on these efforts and on what else you think we should be doing thank you for outlining that Elaine I see Thomas the same thank you Elaine for um putting that all down for the public I said at past meetings that you and staff are definitely engaging in equity work frequently in your in your daily work so I'm glad that you highlighted that for everybody to know um I wonder if you might have considered already out of that news well not new source of funding but reallocating about $50,000 you had said if for you and maybe other members of city leadership were to take um like an official DEI certificate training I looked one up for example at Bentley it's about $1700 for the training um which you know if you add up all of you and leadership that would still be less money than the full-time position at the moment um and would give you some more of that actual knowledge and trained knowledge outside of just you know workshops that you do here and there this would be an actual program um I don't know if that's something that you would consider for yourself and staff uh thanks Thomas I'm here I'm thinking about this for the first time but um off the cuff I'd say that um I think what I what what I keep hearing from experts because I'm not an equity expert what I keep hearing from experts is that most of us spend too much time training and not enough time doing and that said the ideal um the ideal Vermont learning cohort should provide some of that opportunity because also you're right we do need more knowledge otherwise we risk doing harm when we're trying to do when we're trying to act so I'm hoping I would just in this moment I would say that we should rely on those opportunities to get us some more of that knowledge do more of our own peer learning do some more of our own learning um but it to get to the practical pieces we really need to be listening to our community members and we're not hearing from enough of them so that's why I jumped on this idea from the community member of how we can increase that that I would call it porosity between city hall and what's outside city hall and who is outside thank you and I I like I look forward to your advice proposal with this um suggested program because it does feel like it's action when we have done a lot of talking um and I do think that is really important um and listening when we've done a lot of talking to um I have I have a couple things um this might not be decided yet but or something that you can share but I was wondering about who the equity consultant is that school the city and school are working with to decide how to get info on next steps for the equity audit yeah so we um we Paul and I um had an informational discussion with the consultants that facilitated the equity summit in 2019 under you know with the idea that they already have some knowledge of Winooski and they could build on that more easily um Paul also had a really good experience with a previous with a trainer who um is planning to provide a proposal in partnership with another trainer that Winooski has experienced with so it um if you want names I can provide them but um yeah those those are the I'm just providing the reasons why we started with those two awesome that that makes sense to I think that was something I was curious about if we are utilizing current our relationships that we already have instead of seeking out I think I've heard some concerns that it's just seeking out people who aren't necessarily in the community or don't know us as well too and the importance of utilizing people who have already or are either living in Winooski or are already working with Winooski in some way so right actually that's a good point that's a they the people that we're considering don't don't live here so it might be worth another look it it might delay the event that we're thinking but maybe it's you know that seems worthwhile it would be ideal to reinvest the city's money in in local folks yeah yeah and I think yeah next next best is people who've worked with Winooski before that we know or that the community knows um another thing I wanted to note this is both for the community and also fellow counselors one of the items that you listed is the improved access to recreation program registration by reserving half the seats for paper registrations I think it's really important to know and something that Ray noted in the safe healthy connected people meeting is that a huge reason that they're able to do that work is because of the current admin and outreach coordinator which is a position that was originally funded by Essar so this is a huge both connection to the community and potential losing that position then could put us in a lot you know reduce a lot of accessibility and something that has greatly improved access and connection with the city so I think that's really important to note too is the different staff members who are making this possible especially that item which I've heard is really important Elaine I'm curious what the threshold is for putting out like what what the threshold dollar amount is for putting out an RFP for support for consultation work and professional services that is how I posted on my at my desk but I'm going to look at Angela right now it is $10,000 is required for a formal RFP anything over $5,000 we require written quotes okay is that is that a minimum number of quotes at least three quotes if you're just obtaining quotes RFPs it's public RFPs we don't require minimum number of responses yeah great thank you I would say that this list it certainly is not exhaustive I really appreciate compiling it and I I think there's there's a lot of great opportunity and and benefit to having it be a working living document that that we can continue to add items too as as we take on additional action items outlined by the FID audit but I'm also just amplifying the some of the work that otherwise might not go noticed but has significant impact so you know some some of the relational work that's that's hard to quantify is what I'm thinking right and it's I hope that the public heard especially in representation is what's immediately coming to mind I'm sure there's other examples of how any position in community services is an intervention opportunity yes and in terms of the list like I have the the what because currently the the full list on the on the shared drive and just in compiling the was reflecting oh and the fact that Jim you know to your point Jim reminded me that or alerted me that there was a report from the equity summit and I had not been able to find it I hadn't looked very hard so having when you don't have a when me when it's an area that a municipality is not traditionally um uh doesn't have a traditional department with it takes something like this to turn it into a muscle turn it into a habit and having a listing having you know making sure that you're revisiting you know we have things like master plans to remind us of what the goals of the community are we don't have that sort of thing you know built into our DNA yet so something like this is a start to that any questions or comments from members of the public okay I want to make one final note to actually to your final point barrel lane is that we have in those areas under the master plan we have content experts who are helping ensure that we have that data and we're doing that work so part of having that muscle is having a content expert who's able to help the city ensure that this work is being done especially on the city side and as you mentioned quite a number of these items are in part due to the fact we had an equity director in the past so there is that really importance that equity is part of every single position and all the work that the city is doing but there also is that clear benefit of having someone who's able to have that expertise I see in this document as well I would not thank you we move on to item g this is a related item um city council has also been a part of incorporating more equity efforts into the work of the city very brief highlight here in this memo what I wanted to spend a little time on with you all though was first reflecting on the trainings that we recently completed and then having some discussion about what we need to be doing as a council um so I just I reached out to you all a few weeks ago um and didn't hear any sort of consensus things right so I wanted to ask today you know reflecting on the training that we received it was the same one that staff leadership received if there's any like key learnings you walked away with um that you want to call out so for me what I really took away from the two training sessions and um this also came out in the equity audit was the importance of like outcomes over intent and it's something that I want to keep remembering during decision making because I feel like we often do discuss intentions and we don't always know what the outcomes are um so for me that was a I can take away does anyone else want to share there's one thing that I think kind of saw a little bit to today actually with having folks on our leadership team who are able to be remote for various reasons um is thinking about the way traditional structures kind of support white supremacy and limit access and not just white supremacy but other forms of oppression you know it's not intentional so kind of tying on to what you're saying that outcomes versus intent so that's definitely something that was really highlighted in these trainings for me is thinking about those ways of providing access and the ways that some kind of tied to that too is almost building community and trust to be able to um hold those spaces and to have that accountability and understand and guess you know understand where that people are coming from these different places and that we want to make spaces um that everyone can be able to show up uh and fully participate as themselves for me I think um I tend to take the action oriented um takeaways where um you know council heard as already mentioned there's there's a lot of talking but I think um you know the the summary memo that Elaine put together shows the action behind the talking it shows the um the dedicated effort that city staff um and council are putting into um you know really exercising that um effort that muscle that Elaine just mentioned um of making this part of our day-to-day and and that to me um I think having tangible um actionable items um is is to me what stands out most from those two sessions. So next um I think that it looks like council is about to be sorry Thomas um I I think it was a helpful refresher on some of the ways in which um especially our approach to work and interaction has continued can get wrapped up in white supremacy culture concepts of quantity over quality and uh valuing certain forms of input sort of methods of providing thoughts and opinions about things that we tend to shy away from emotion we want written information so those are good refreshing our refreshers of those types of um just some considerations that have been going into decision making like oh well what should we require any written what should we require be said and how should we respond to difficult comments directed at us I think those are really important things to remember it's it's not something you have to protect yourself from every time especially in public space or position so I think that was helpful to have that reminder and what I took away kind of the place I got to reflect a lot during that training was around um the ways that I listen and then the ways that I'll speak or um try and summarize or reflect back what I'm hearing um and using facts as opposed to impressions and uh listening or the the underlying like why is this important to you why are you raising this issue now why are you saying it this way and does that help me something more about what I'm might otherwise and I said by focus only on the words so I think that was really helpful to have some practice with that and some reminders or some new ways of kind of thinking about those communication skills um yeah I think I kind of just echo a lot of what everybody said I've done so many of these trainings that I feel like what I I take away from most of them is really trying to and the importance of understanding that everybody comes from a completely different background even if you think that their background is similar to yours it may not be at all and then the slightest and that forms the way that people come to a conversation or um just just day to day living and to to understand that as you go into something um and one of the things I mentioned to the mayor is that I think we should do more of these types of trainings no one organization or trainer um and I think the ones that we had came to us that they don't know everything there's no way for them to know everything and this world of DEI and everything that's coming out and evolving um so personally um I'm I'm totally down to do more of them um I don't know about everybody else but um just just keeping an open mind I think is is one of the most crucial things I learned from those and I'm reminded well that's a good segue into the second portion of um the discussion which is potential next steps um and and we're not here to like vote on anything or make decisions that impact budget operations um but I wanted to do some just get a sense of like how so we just saw like a bulleted list of specific actions that staff are taking to incorporate equity into operations and I want us to think about how we incorporate it into policy and community engagement separate from that right so like what what can we do that doesn't involve operations or like staff and financial changes um you know for example I think the community engagement strategy we developed is probably the most important thing we can we have and can do to make more of those connections um I think it's a little loose right now we can get tighter on um actually putting that into practice um and I'm curious what other you know another suggestion I had is um an activity we did at my the organization I worked for was adopting shared statements on what does inclusion mean what does equity mean um to try to kind of ground us together make sure we're aligned so opening it up to to other ideas I like the idea of kind of developing those shared definitions and so put forward maybe thinking about even diversity and belonging I think it's belonging in particular because that's something that we're using a lot um but I think it's even valuable to look at diversity which might seem to have a very set definition but thinking about it in the context of the particular work that we are doing I think can be really important um seen some example when I'm like thinking like how higher ed thinks about diversity stuff like that so I definitely do see the value in that of having a grounding document to come back to as well so I'm thinking about putting in words what we're thinking about so we can ground ourselves as well as kind of be transparent to the community about what we're doing um I think something that you know council has been doing um at least this past year from what I've observed but um should continue to do and and consider expanding on this um you know just making sure that we're attending the public um the the various events and celebrations around the city not just the events that downtime with new ski holds but um things at the school things at the senior center things at the brine center etc um I think those those are areas where um just having greater like we have our own city calendar and links to the school calendar but if um if there are other events that that we hear or know about that um that we can participate in um I know that there have been a couple uh block parties for example and and just like encouraging more of those participating in those um I think that that helps create additional trust and familiarity and is something that's doesn't put additional resources on staff or impacts the operational board. Living with the O'Brien Center directly behind me I definitely see there seems to be some more community gatherings there and I don't know if there'd be a way to possibly get invites I definitely wouldn't just want to show up. Sure of course. Yeah I'm not suggesting no no no I understand I was thinking for those in particular um I think that that's that's a really great point um and some of that might be then making more connections that we can bring back into the community engagement too. I um one item that I would love to see I mean maybe this could come along with the development of a shared statement but I think it would prove us to look at some sort of auditing tool that we can easily and quickly apply to major policy decisions for example we had our discussion around eviction protections and general protections more broadly like there could be some place for a somewhat um I don't want to say objective but a way for us to structure our thinking about how policy is intersected with equity we don't really have that right now but we might have that conversation ad hoc in a given context I think there are templates out there and we don't need to have something perfect but we could have something that's good enough for us to start more structuring our review policy because that's something that can last that outlast any individual counselor's training and tenure on council and I think that's in all of these I would love to see what we can have here that could potentially still be on the website if all of us were no longer on council as a thing that we do right as part of our policies and procedures our standard operating procedure as council that we use these tools and certain steps like community engagement plan or strategy and the like a policy audit tool so those are things that I think would be more I would like to see us direct our efforts there's some of those things that can be institutionalized beyond individuals and that's an I that's one idea I have that seems to be um could outlast us I have collected several of those but I can bring to a future meeting at some point I wonder too if that might be maybe use it use some of those templates but I wonder if that might be even something that Alina and I could bring to the folks that ideal and see is there a Vermont municipal specific one that could be generated so we're definitely looking to bring those things that the LCT is working on but um yeah I think there is actually one used by the state which a couple of municipalities have adopted so yeah all of the above all options I do think it's one of those there's there's likely to be little harm in adopting something imperfect at this point um and and potential gains even if they're not fully as much as we could do we can still make easy effort and I think that's also important to not get hung up on waiting till we have a perfect but to be able to make progress without harm as you were saying the only other feedback Al shares I feel compelled to say this because it has been weighing on me more over the past months is as a parent is it's hard to attend more events um so I think that's something that for me I would like to encourage us to think about what is a reasonable expectation of the counselor especially if we want to have a council that represents the range of community members of the city people with kids and people with kids of in ages between five and eighteen or requirement different type of work and commitment and and attention and I think that's something we gotta keep in mind um that the the ability to participate outside of these meetings is variable for folks in our community as we discuss like Saturday morning being better uh for some folks and then um for people who might consider running for council I think we want to make sure we're setting ourselves up to still be for these positions to be um accessible to as many people as possible in the community I couldn't empathize harder with that not as a parent but like the volume of commitments that we make and I have in the past like I don't know approach it from a more selfish lens more just like I am over committed but you make a very good point like as we as we set up new structures commissions bodies events we need to attend who realists is that shedding people out of of doing public service I think it's a good question to to keep in mind I think there's uh some clarification then that would be beneficial of like an expectation versus something that does benefit and create more trust um and correlates to creating and building more trust and familiarity like I don't I think it it should be um capacity should be part of that conversation of course um and that attending events certainly is a way that it doesn't you know it doesn't um hinder staff capacity or operational needs obviously it factors into council needs council capacity I should say so I think that for me that recommendation is like within within a councilor's capacity yeah I just I think this goes into formalizing this community engagement strategy further having more structure to it and having a bigger picture of what we are all doing or asking ourselves to do and certainly some people have more time for certain things than others but yeah having some visibility into that new council orientation like here's sort of the you know we went through a list of um we did the liaison assignments right and then the airport commission came up but we created another commission um so we just we keep building on that and and we should reflect on what the big picture is one thing that you know hate to bring up with the budget and this feels so far out of reach is and is kind of awkward for a standing council to vote on is but considering young future is compensating the council in a way that is more accessible to people because of the time commitment similar to talking about compensating our commissioners I think that's something to acknowledge too was this is it's a lot of work for little pay and that uh leaves out a huge part makes it very accessible to only certain members of the population who are able to do that which again awkward but I think it depends on what you do this as a paid position or volunteer with stipend I think that's I think that's a discussion we're having but I think to try and maneuvering people for the actual hours spent in service of the as a different model and it's a it's a model that I don't know our community can support right now no no that comes with it yeah but I think it's can be good to acknowledge that there is a cost barrier even if it's not something that is super obvious to doing this work I had one other note but might tie a little bit to what council runner was bringing up to and thinking about ways to kind of continue our own learning and acknowledge that kind of all equity work requires kind of constant no one's ever going to be an expert we're always going to need to learn and I wonder if there's a way to and here's adding more work so um how this might work but adding a way to do something similar to what kind of staff leadership is doing and kind of uh like have a rotating um one counselor maybe once a month brings in I don't know if this would be in a meeting but somehow brings forth a reading or an opportunity um to kind of do more equity learning um in some way so just kind of building that into an expectation um and also um taking hopefully again putting that on the council instead of putting a burden on city staff noting all these ideas right we were not committing to anything yet of course we'll have a follow-up conversation I wonder if it could in some ways also intersect with our liaison appointments like if we could focus our like part of shared level learning that we try and share at back is some expertise we then develop in our commissions as well like I could see that that there's like that that co-benefit right like it's not just the equity benefit but also the benefits of the work yeah that the commission and the council is trying to accomplish in the policy areas um it may be helps founded not founded but like make it more um make like attackable because what is the world of equity I'm going to take into account the list month and like I don't know that I'm not an expert but if I talk about housing and equity I feel like I could find some some folks that I would find motivating to learn about and then share back I think that could be a way to to structure that oh an interesting way to to co-structure that yeah yeah and I think looks like planning and municipal infrastructure and economic vitality those are areas where it's really interesting to explore how equity very much does intersect I think in ways that sometimes it's less off less obvious than say say how we connect to people well unless there are any other burning ideas it's 10 p.m. and we have an executive session that one more yes you can maybe we added goal update for council and community engagement strategy we did kind of have benchmarks and goals listed in there that could be a monetary outstanding monitoring item that we put ourselves to um and even if the answer is well we haven't done anything it at least keeps us honest to that strategy because I didn't think you're right Christine may have that um being engaged in a strategy by some of these some of these structures that we might want already and this could be a good way to monitor it excellent okay thank you oh well last call for public comment any thoughts questions mario did try to raise her hand oh i didn't see it always like into the next comment but i don't want to call it out specifically mario i know it's 10 but if you did have something okay bring her over paul thank you can you hear me yes welcome thanks um i was just going to mention that i think blaine intense at reiby in burlington had made one of those policy added tools that is being used um for burlington it is pretty anyways you can ask her for a copy of it and if you need her email i'd be happy to share that but it sounded like you already had a collection of them there so maybe you don't need them um but i'll add since i'm unmuted that i appreciated having that hearing that conversation live in front of me it's just thank you for being candid um and speaking about about your equity goals and sharing that with the public i appreciate it thanks mario i'll follow up because i would love another example but okay all right so this is the end of our regular agenda we've worn an executive session um i'm looking for a motion to find that pursuant to state statute section 3131 e regarding civil litigation or prosecution we should have this conversation in an executive session so it was not to put us the city at risk so second motion by jim second by brin all those in favor please say aye i'm sorry i'm losing it all those in favor please say aye aye motion carries um am i inviting just elaine and angela i don't think we need it angela oh just you elaine yes oh good for you angela okay um so i'm looking for a motion to move into executive session inviting the city manager elaine wong so moved motion by jim second by brin this time all those in favor please say aye motion carries um we have warned an action coming out of this executive session so we'll move into executive session we're only going to discuss this single topic no other business will be discussed when we come out of executive session we may take a vote on on the remaining action item to adopt a settlement resolution following that the meeting will adjourn we are on item 10a on for discussion approval adoption of the settlement resolution do i have a motion to approve so moved second motion by jim second by thomas all those in favor please say aye aye motion carries this brings us to the end of the agenda do i have a motion to adjourn so moved second motion by jim second by aurora all those in favor please say aye aye motion carries meeting adjourned good night everyone


In the previous video we learned how to find a basis for the column space of a matrix. Now in this video I want to talk about finding a basis for the null space of the matrix. This one's a little bit more involved. Remember that the null space of a matrix A, this is going to be the solution set, the solution set of the homogeneous system AX equals 0. And so we have to basically solve the problem, solve the homogeneous system in order to find the solution set. And then when we do that it turns out in the process of solving for the solution set we can actually pull out a basis. We've been doing it already. We just didn't know it. Let's explain why it works. Now what they remember is that if two matrices are row equivalent, what does that mean? It means that you can transform one matrix into the other by some sequence of row operations, replacement, interchange, or scaling. And if you have a system of equations like AX equals 0, replacing a matrix with a row equivalent matrix doesn't change the solution set whatsoever. And as a consequence, if row equivalency doesn't change the solution set to a linear system of equations and the null space is a solution set to the homogeneous system, that means if A and B are row equivalent, then the two matrices will have the same null space. The null space of A is equal to the null space of B. This is particularly useful if B, for example, is the RREF of the original matrix A. So our strategy for finding a basis for the null space is going to be the following. We're going to use the same technique we've done before to solve the homogeneous system like we did for example in section 2.6. And then we're going to focus on the non-pivot columns of the matrix. These non-pivot columns are going to correspond to the free variables and the free variables are the things that are going to produce non-pivot solutions to the homogeneous system. So that's how we're going to proceed. I'm going to use an example to explain what's going on here. Consider the following three by five matrix A. We're going to construct a basis for the null space of this thing. Now if you're paying attention, this matrix might seem familiar. Turns out we played with this exact matrix in the previous video of this series. So we showed that if you take this matrix A and you perform some raw operations, you can get the following echelon form of the matrix. Now that matrix is not yet in row reduced echelon form to solve the null space here. Although you can get away with it in echelon form, you probably want row reduced. And so let's do one more step to get rid of the three right here. We're just going to take row one minus three times row two and we get the following matrix right here. So this matrix right here is the r, r, e, f of A. Now we are solving a homogeneous system. So be aware that I did augment the column zero here. But the fact that you have a column of zeros means that when you do row operations, nothing will ever change. So when we did, when we found the basis for the column space, I didn't put this, this column of zeros on here. That was okay. I don't need to do that because I'm doing the same raw operations, but they don't affect this zero column. So therefore, I can slap it on without any big deal whatsoever. The reason I'm including it this time is to emphasize that this augmented matrix represents a system of equations. So when we take the r, r, e, f form of this augmented matrix, the first row will give us x one plus three x three plus five x four plus x five equals zero. The second equate, the second row gives us the equation x two minus four minus x four plus x five equals zero. Now the third row actually is just the equation zero equals zero, which is offering no benefit to the system whatsoever. So I omitted it here. Now, because these, you're looking at these, these equations here, we see that because of the pivots, these are going to be dependent variables. And so like we saw before, the first and second columns of A give us a column space for the, they give us a basis for the column space. But for the null space, we're going to focus now on the non pivot columns, because these are what's going to give us the free variables. The solution to a homogeneous system, its size is dependent on how many free variables you have here. That's why the nullity is going to be three here, accounting the free variables. And so if we solve the dependent variables with respect to the free variables, we get that x one equals negative three x three minus five x four minus x five. Notice that the coefficients of x three, x four, x five changed their signs as they moved to the side of the equation. And then we also see that x two is going to equal x four minus x five. So again, the coefficients of x four and x five changed from positive negative or vice versa when we move to the side of the equation. That's an important detail that will be helpful in a little bit. Just kind of notice that. So we saw this system, we have these free variables in play here. So if we look at the general solution to this homogeneous system, this is the vector x, which has five variables, five components, x one, x two, x three, x four, and x five. Now what we've learned here is the following will x three is a free variable. So it could be whatever it's wants. And you know, same thing with x four and x five, you know, they're just like queen, whatever I want to be x one and x two on the hand, they're dependent variables. And so we have to use these assignments from above x one will be negative three x three minus five x four minus x five and x two is given as x four minus x five. So this is the general solution. But this general solution, I could decompose it into smaller vectors. That is, I could, I could kind of rip it apart into the three vectors corresponded to the three, the three free variables. That sounds like a Dr. Seuss thing right there. Three, three, I can't even say it, three free variables. I need to go read Fox and Sox tonight, I think to my kids to practice this for the next lecture. So if you take the first, take the first vector associated to x three, so look at those terms in this vector that involve an x three like so. So we get the vector negative three x three, zero, the second one didn't have any x threes in it, we have an x three, and then the last two didn't have any x threes in it. The next one, we're going to have a vector for x four, which there are a few of those inside the vector right here. So the next one would look like negative five x four, x four, zero, x four, and zero. And then lastly, we're going to do a third vector for x five, we're going to do this for each of the three variables x five, x five, x five, like so. So make sure you grab all of those. This is going to give us negative x five, negative x five, zero, zero and x five. So we separate, we separate our general solution into a sum of three vectors depending on the three free variables. Oh, see, I practiced that one. And so for each of the free variable vectors, factor out one, factor out the free variable. So for the first one, you can take on an x three, and this gives us negative three, zero, one, zero, zero, like so. For the next one, you're going to take out the x four, and that leaves behind a negative five, a one, a zero, a one and a zero. And then for the last one, take out the x five, and that gives you negative one, negative one, zero, zero and one. And so for clarification, or maybe just simplification, let's call this first one you, the second one V, and this third one W. I think that's the alphabet. And so what we've now see is that this right here, we're saying that x is just the combination x three times U plus x four times V plus x five times W. And since x three, four and five are free variables, we can choose them whatever we want, x will be a linear combination of these three variables right here. And as x was the general solution, this then gives us that the null space of A is equal to the span of these three variables, U, V and W. And that agrees with what we said earlier, the nullity of the matrix is how many, how many free variables are in the system. We counted there was three, and therefore it should be span, we should be able to span it using three independent variables and three independent vectors, I should say. And we have in fact our three vectors. But how do we know they're independent? Look at these three vectors and I'm going to make the following kind of argument. If we look at just, look at just the third position right here, you'll notice that every other vector has a zero there. And so if this thing is going to combine to give me an x three somehow, I basically got a fourth x three to be this coefficient, because no one else is going to contribute to the third component. So there's no way that this thing could add up to be zero without this coefficient being zero, not going to happen. And same thing for the other ones, right? If we focus on the fourth entry, notice here that in the fourth component, V has a one, U and W have a zero. If this were to add up to the zero vector, it would basically force that this coefficient have to be zero. And again, same thing for the fifth component as well. The fifth one W, that is, I should, the fourth, the fifth component of W has a one for you and V, they're both zero. The only way you could get a zero in the last component is if we had a coefficient right here. So these vectors are independent. Basically, if you kind of forgot, we put this together as a matrix and you kind of forgot the first two rows, this would be a matrix in row reduced echelon form. That's going to be independent. So we do have, we do in fact have an independent set of vectors which span the null space. So this right here gives us a basis for the null space. And that's really great. And this actually the technique we saw before, right? So in the process trying to solve homogeneous systems, we actually were finding bases for the null space. But it turns out we can dramatically simplify this process. How are we going to do that? Well, I'm going to come back up to the original matrix right here. Now, one thing I should mention in contrast with the column space, the column space, the basis will consist of vectors from the original columns of A. When it comes to the basis for the null space, these vectors do not coincide with columns or rows of A. They do not coincide with rows or columns of the RREF either. They come about from solving the system of equations. But what I want to mention to you is that this process of finding the RREF, pulling out a general solution and ripping it apart, this process can be dramatically simplified in the following manner. So when you have your RREF and you want to find the null space here, what you need to do is you need to identify who are the free variables which we saw before. And you're going to create a vector for each of the free variables. So there is a vector that's going to be associated to X3. So what you're going to do is that you're going to just put a little asterisk right here in those positions that are corresponding to pivot positions. And then for the free variables, you're either going to put a one or a zero dependent upon RREF on X3. You put a one and three spot, you put zero and four and five. Then there's going to be a free, the free variable X4 will produce a vector that gives us the basis. You're going to put a star, a star where the dependent variables go. You're going to put a one in the fourth position and then zero in the other free spots. And then lastly, there should be a vector associated to X5. You're going to put a one in it in the fifth position, zero in the indices that correspond to the other free variables and then put asterisk, just these little stars in the one and two spot. The reason we put stars for one and two is that for one and two, those were the dependent variables. So it depends on which of these variables we're looking at. So you start off with these ones and zeros. Okay. Now what you're going to do is you're going to fill in these stars. So let's look at the first row here. These stars are going to coincide by looking at the first pivot row. So we know that X1 is a dependent variable because there's a, there's a pivot position in the first column, but then associated to that first column, there's a pivot row. We're going to read off the numbers in this pivot row corresponding to it. So if we look at the first row, notice in the three spot, you have a three. And so I'm going to record a negative three right here. In the fourth column, we have a five. I'm going to write a negative five right here. And then here in the fifth column, we have a, we have a one, I'm going to write a negative one right here. So we're going to write down the numbers we see in this first column and this first row, excuse me, based upon which variable is it connected to? Now you might wonder why did you do a negative three as opposed to a positive three? Remember when I mentioned earlier that when you move the, when you move the three X three to the other side of the equation becomes negative, that is what we're incorporating right here. So as you go from the matrix to the vector, it's like you're moving the variable to the other side of the equation. It switches its signs. How about X2? How about X2 here? So you look at the second row, you can ignore the pivot columns, but we look at the second pivot row. So in the three spot, you get a zero. So we get negative zero, which is still zero in the fourth spot, we have a negative one. So I'm going to record a one. And then in the fifth spot, we have a one. So I'm going to record a negative one. And so there, this right here will be our basis for the null space of the matrix. And hopefully these vectors look a little bit familiar, right? So the first one was negative three, zero, one, zero. I wonder if I can get them all on one slide here. I'll have to zoom out a little bit. You can then compare, right? Negative three, zero, one, zero, zero. That was the same thing. The next one was negative five, one, zero, one, zero. Same thing. And the last one's negative one, negative one, zero, zero, one, same thing. So we can actually extrapolate these, the basis of the null space directly from the RREF. And we can kind of skip all of this middle stuff with systems of equations, general solutions, if we want to. Now, if this more long drawn out process makes sense to you, that's fine. But I did want to show you this nice little shortcut. In the next video, I'll do another example of the shortcut method for finding the null space of the matrix.