Hello everybody hooded Cobra Commander 788 here and I'm back with another vintage Cobra toy review for the third week of Cobra Month and this time we're going to do an army builder. We really needed to do an army builder for Cobra Month, really Cobra had some great army builders and we have a really great army builder for this review. We're going to look at the 1985 Cobra Polar Assault Trooper the Snow Serpent. This is the Snow Serpent Cobra's Polar Assault Trooper. He was first introduced in 1985, he was also sold in 1986. He was discontinued in 1987 and in 1987 we did get another Cobra Arctic Trooper the Ice Viper as a vehicle driver. The Ice Viper was the driver of the Wolf Arctic Attack Vehicle. In 1992 the head of the Snow Serpent was reused for the Heli Viper, which is one of the ugliest action figures I have ever seen, but sadly not the ugliest. The Snow Serpent was Cobra's first cold weather specialist, but by this time GI Joe already had several Arctic Troopers, the first of which being Snow Job from 1983. In 1985 we got Frostbite as a vehicle driver, the driver of the Snowcat, then in 1986 we got Iceberg. Let's take a look at the Snow Serpent's accessories starting with his weapon and This is an AK-47 with a folding stock molded in grey plastic. The AK-47 was a Soviet assault rifle developed by Mikhail Kalashnikov. It's an appropriate weapon and a very nice representation of that real-world weapon. And I like the folding stock, that's a nice difference from other AK-47's we got from GI Joe figures. The 1982 and 83 Cobra officer also came with an AK-47, but with a fixed stock rather than the folding stock and in a darker grey color. Around his waist he had a blue pack that is listed on the card contents as a parachute pack.  This is a pretty nicely detailed accessory. It loops around and latches through this buckle here around his waist. It's a nice blue color and it's a nice, kind of a nice color that breaks up all the grey.  Nice color and very nice detail. What this accessory means is that not only are the Snow Serpent's Arctic troopers, they are also paratroopers. The parachute pack is molded out of a softer plastic, it's very pliable. You do run across some of them with broken straps sometimes, but they're pretty robust and usually you can find them intact. Attached to the side of his backpack is another weapon, his missile pod, this missile launcher, which connects to the backpack by this peg and this slot here and that is very convenient. I wish more accessories would work this way where we had a place to store them, so that is a nice bonus, I like that very much. The contents of the card on which the action figure was packaged call this an anti-tank EK99 missile and I don't think this is a copy of a real world anti-tank missile system. Normally you don't see anti-tank missiles that are fired like this with the missile tip sticking out. This may be loosely based on the M47 Dragon or the FGM-148 Javelin, but it's not an exact replica of either of those missiles, this is kind of a made up thing, but it still does look pretty good and it means that the Snow Serpent is also an anti-tank missile specialist. The missile came with a folding stand and with the stand you could stand it up like that, looks pretty good and the Snow Serpent can stand behind it and fire it at GI Joe tanks. This stand does come off, so you will often find this missing, you will see a lot of these anti-tank missiles with the stand gone, so that's something you'll need to look for. Even though we've looked at a lot of accessories, we're not done yet, he has more. He has his backpack, which pegs into his back like any other GI Joe backpack did and this is a very large backpack, it has some nice sculpting and detail on it. Like I said, it does have this tab on here to attach the anti-tank missile, you can just tab it in like that, push it in and it holds on pretty well. This is a very nice feature, I like this a lot, this is great. This is a nice looking backpack, it's large because the Snow Serpent in the Arctic environment is going to need a lot of survival gear, so this is appropriate, great looking backpack and finally the Snow Serpent has a pair of Snow Shoes. This is pretty cool, he has two of them of course, one for each foot, they peg into the feet by this hole, it has a foot peg there, that's how it attaches to the feet. They are both the same, there's not a separate left and a right one, so you can fit them on either feet and it does fit quite well, quite snugly and it's reasonably well detailed. This is a nice change, a lot of the Arctic troopers that we got especially from GI Joe, they had skis, they came with skis, so we got a lot of skis in GI Joe, so this is a nice alternative to the skis that we usually got. The Snow Shoes work pretty well as a figure stand, with the Snow Shoes on you don't really need a figure stand for the Snow Serpent, in fact they're pretty stable and you can even get them in different poses with these Snow Shoes. Out of curiosity I wanted to see if Snow Jobs skis would fit on the Snow Serpent, just to see if maybe you could have them as an alternative to the Snow Shoes if you wanted the Snow Serpent on skis, so we're just going to test that out right now and it looks like no, they do not fit, the Snow Serpent's foot just goes a little bit too far for this front little notch here where the foot goes, Snow Jobs' foot fits in there pretty well, but the Snow Serpent's foot will not fit that ski. Let's take a look at the articulation on the Snow Serpent, he had the standard articulation for 1985 GI Joe action figures, that means he could turn his head from left to right, he could also look up and down, he could move his arm up at the shoulder, his upward movement might be slightly hindered by the sculpting of the fur line collar there, he could swivel his arm all the way around and that definitely is obstructed by that fur line collar, he could bend at the elbow about 90 degrees, he had a swivel at the bicep, he could swivel his arm all the way around, the figure was held together with a rubber O-ring that looped around the inside so he could move at the torso a little bit, he could move his legs apart about so far, he could move his leg at the hip about 90 degrees and he could bend at the knee about 90 degrees. Let's take a look at the sculpt design and color of the Snow Serpent, starting with his head and on his head he has a light gray helmet, you can see that goes all the way around, some ridges on the top of his helmet, he has blue goggles and he has a black face mask, almost looks kind of like a hockey mask or something like that, this face mask has a nose that kind of sticks pretty far out there and the paint on that nose can wear off, that's probably the most frequent paint wear that you'll find on this figure, so take a look at the nose and if you want one that's in great condition, just check to see if any of the paint has rubbed off there, this helmet and mask covers the entire head and face which makes this figure perfect for army building, you can't see any features, there's no individuality to it so you could get a lot of these and put them together and pretend you have an entire army of Snow Serpents, this really is the perfect army building figure, on his chest we see he has a fur lined collar and that goes over his shoulders here and the sculpting on the chest actually goes over the arms a little bit and that's white on top of a light gray winter coat and this is a nice subtle color difference between the white collar and the gray coat and it's kind of nice I think in these early G.I. Joe action figures now that they can have some subtlety like this they didn't all have to be high contrast and neon colors, he has some blue straps and that looks like it's actually a harness that would go to his backpack and his parachute and that blue is a nice little splash of color against the light gray that looks really good continues around to the back and this is a highly detailed chest piece here this is a lot of detail on this winter coat that he's wearing you can see a zipper there he's got some seams and some pockets he's got some buckles on his straps on his harness it even has a strap here with a little buckle on both sides I'm a little bit torn though because these are unpainted details and as you know I'm not very fond of unpainted details it would be nice to have these sculpted details painted in however I think that with this color scheme adding an additional paint application or color there would be too much I think that would make the figure look too busy so on the one hand I like the detail and I don't think that it should have been painted but I do prefer painted details so I'm a bit on a bit of a quandary here I like it in some ways and I don't like it in other ways on his right arm he has this pouch here and on both wrists he has these gloves these furry gloves and what's interesting is the fur doesn't go all the way around it almost looks like he has like white Sasquatch arms or something like that it looks like he has furry arms on his left arm we have a nice red Cobra sigil and that's another nice little spot of color on this otherwise you know very muted colored figure the waist piece continues the theme we've got a zipper here we've got the blue harness and belt that continues around to the back we've got some nice sculpted detail on the trousers and this may in fact actually be a jumpsuit I think that's what this is this supposed to be a jumpsuit with a blue harness over it the legs are also done in this light gray plastic with a couple large pouches here on the front and there's some nice detailing along the sides looks really good but it's also kind of plain he doesn't have any weapons sculpted on his legs no pistols no daggers nothing like that no grenades but that's fine I really think that if they had done too much down here it would have been a little bit too busy there's a lot of detail on this figure already and you don't want to overdo it and then we have some blue boots that match the color of the harness let's take a look at the file card the file card was printed on the back of the card on which the action figure was packaged you can see some of the artwork from the front of the card there it shows its faction as cobra obviously and it has a portrait of the snow serpent here his specialty is cobra polar assault codename snow serpent it says his file name is classified but I don't really think it should have an entry here for the file name we're not talking about an individual snow serpent here there were a lot of snow serpents in cobra forces and they all pretty much look the same so I don't think it should say it should say file name classified here it's not that his file name is classified but these guys would have a lot of different names primary military specialty is arctic operations secondary military specialty is infantry birthplace various countries this section says snow serpents are the arctic specialist branch of the eels in parentheses cobra frogman when the file card refers to eels they are talking about these guys the cobra frogman that were also introduced in 1985 the same year as the snow serpent they must undergo the same rigorous training program as the eels with the addition of a six month cold weather course somewhere above the arctic circle so if cobra eels are elite soldiers within cobra snow serpents are the elite of the elite other aspects of their training include airborne operations in parentheses under arctic conditions anti-tank procedures and the use of snowshoes skis and kayaks I like the way this file card references all of the accessories that the snow serpent comes with talks about anti-tank procedures and he comes with an anti-tank missile it talks about airborne operations and he comes with a parachute pack that's a nice coordination between the action figure and the file card this bottom section has a quote it says if the eels are the elite of cobra's naval branch then the snow serpents are the best of the best how else could you characterize an individual who would parachute onto an ice flow in sub-zero temperatures and then be prepared to march 50 miles with full field pack assault rifle and anti-tank weapon so in the hierarchy of cobra special forces the eels are the elite and the snow serpents come from the ranks of the eels and are a step up from the eels in the comic book miniseries gijo order of battle published by marvel comics issue number three the snow serpent is featured on page 19 with a really nice drawing of the snow serpent the information here is the same as on his file card he seems to have a different missile pod this is probably based on one of the concept drawings from before the figure was released and also you can see he has his snowshoes stored here on his back and that's a feature that the figure did not have it would have been nice to have that that would have been a great feature that may be the one downside to all these fantastic accessories is that if the snowshoes are not on his feet he doesn't have anywhere else to store him he can store his missile pod but not the snowshoes so I think maybe a couple pegs on the backpack or something for the snowshoes would have been a nice bonus the snow serpent was among a set of fantastic cobra army builders that came out in 1985 a 1984 may have been the best year for cobra but starting in 1985 we got to get a lot of really great army builders we had in 1985 the cobra eels the frogman we had the elite trooper the crimson guard of course the snow serpent we had the cobra tele viper which was the communications specialist we even had a vehicle driver the lamprey who was the driver of the cobra moray hydrofoil oddly enough the basic viper figure the cobra infantry trooper did not come out until the following year 1986 taking a look at the snow serpent overall this is a figure that is extremely well detailed he has so many details that some of the details are unpainted which normally is something that I do not like but it is almost forgivable on this action figure the choice of colors on this figure is superb the obvious color scheme on a figure like this would have been all white kind of like snow job but they didn't go all white for this figure they gave us some nice light gray with a few other colors thrown in this is add some color interest to this figure which could have been very plain and boring the snow serpent is loaded with accessories and they are all appropriate accessories for his specialty there's nothing weird here these are the exact accessories you would want to get with a snow serpent even seems like you get some bonus accessories with the missile pod and being able to connect the missile pod to the backpack is great they really went the extra mile with this figure's accessories it's even a pretty robust figure there are not a lot of common points of breakage on this figure of course like any vintage action figure you'll find some with broken thumbs and broken crotches but overall this is a pretty tough figure you don't run into too many broken ones I think the main thing you need to worry about is probably paint wear and really you mainly need to worry about that on his nose that black paint tends to rub off but even the paint wear isn't really a major issue you will often find them missing accessories and I guess that's the downside to having so many great accessories is a lot of them do tend to go missing so you might have a little bit of trouble tracking down accessories but it's worth it they are really great accessories I consider this to be a top tier cobra figure and it's really hard for me to think of any arguments against putting him in the top tier he's not a major character like Destro or the Baroness but as far as the sheer quality of the figure goes this is top notch and I'm saying this as a person who didn't really play with a lot of Arctic figures as a kid where I grew up we didn't get a lot of snow so we did not play with our toys out in the snow very often so I don't have that kind of personal connection to this figure but looking at this figure now I have to admire it as far as snow serpents in the G.I. Joe media I don't really have a lot to show you they did show up every once in a while a few times in the animated series and I think we saw them a few times in the comic book but they were never major players so really not a lot to show you as far as media goes the snow serpent was perhaps underutilized in G.I. Joe media he maybe deserved the spotlight a little bit more that was my review of the 1985 cobra snow serpent I hope you enjoyed it and we are reaching the end of cobra month we have one more video review to do this month and we are done with cobra month I hope you've been enjoying these cobra review videos and I hope you'll stick around after cobra month I've got a great lineup of toys to review next month you don't want to miss those so make sure you subscribe to the channel on YouTube make sure you like this video if you did like it on YouTube and don't forget to like the Facebook page I've got a lot of updates there you do not get anywhere else thanks for watching I'll see you next week with another vintage cobra toy review see you then


See one of the central notions in linear algebra is that of a linear transformation this is also central not only linear algebra but the entire mathematics. Today I will discuss the notion of linear transformation give several examples now these examples will justify the statement that I made just now you will see that linear transformations arise in differential equations in integral calculus in matrices for transformations between vector spaces etc. So let us first look at the notion of a linear transformation look at several examples and probably towards the end of the class we will look at some properties some simple properties some not so simple properties and then we will be able to compare how a linear transformation behaves with a general function between vector spaces okay. So let us begin with the notion of a linear transformation between two vector spaces. So I have two vector spaces let us call T as a linear transformation what this means is that V and W are vector spaces T is a function to begin with so I am just emphasizing it is a function that satisfies the following two conditions T of U plus V equals T of U plus T of V for all U V in V and T of alpha U equals alpha T of U this is true for all alpha in the underlying field which we are assuming is a real field and for all U in V. So a linear transformation is a function linear transformation between two vector spaces is a function between those two vector spaces that must satisfy these two conditions let us observe that the right hand side vector is in W what is inside U plus V that is in V T of that is in W and the formula for T of U plus V is given by this the formula for T of alpha U is given by this right hand side okay. Sometimes we write T U instead of T of U okay this is just a notational convenience. So let us now look at examples let us dispose of the trivial examples first one you look at the map which I will denote by 0 0 from V to W defined by 0 of U is 0 for all U in V see the left hand side 0 or O is a 0 transformation the right hand side 0 is a 0 vector in W okay trivially this is a linear transformation because it satisfies these two conditions okay this is called the 0 transformation example 2 I will call it I I from V to V you know what it is the identity transformation defined by I of U equals U for all U in V now you see that the right hand side U is the same as a left hand side U that you started with so you need W to be equal to V for identity transformation W is equal to V so these two are trivial examples of linear transformations let us now look at one non-trivial example and then probably two examples from geometry motivated by notions from geometry.  So these two examples come from geometry let us also look at some other examples coming really from geometry but this time we may have to look at higher dimensions that is my fifth example let me look at T from R m to R n let me take the case when m is greater than or equal to n I will define T X 1 X 2 etc X m m is greater than or equal to n that is n is less than or equal to m so the number of coordinates on the this on this right hand side I have a vector which has less than m coordinates there is a natural definition X 1 X 2 etc X n X is in R m so what I have dropped is what I have done is to drop the coordinates from n plus 1 to m okay this is called a natural projection T is called a natural projection on R m now you can verify that this T is linear these are some of the simplest examples of linear transformations this T is linear let us look at the usual projections projections that we encounter in engineering drawing for instance T from R 3 to R 3 defined by T of X 1 X 2 X 3 this let me say it is X 1 X 2 0 X 1 X 2 0 these would be called projection operators T is called a projection operator on R 3 we will reserve the word operator when the vector spaces V and V are the V and W are the same if V is equal to W then linear transformation will be in particular called a linear operator so this is an operator it is called a projection operator you see that any point on the plane rather any point in 3 space is dropped on to the horizontal plane the plane let us say X Y plane any point in the X Y Z plane the Z coordinate is 0 so we are looking at the projection of any point in 3 space on the so called X Y plane that is the projection operator this is just one of those examples I have another several other examples for instance T of X 1 X 2 X 3 could be X 1 0 X 3 or X 1 0 0 on the X axis etc all these are called projection operators let us look at other example let us take this time T from R m to R n with M less than n it can be equal also the definition is as follows this will be X 1 X 2 etc X n if it is strictly less than n the other coordinates are taken to be 0 so these are n minus m components m is less than or equal to n so this may not be there then it will reduce to the natural projections but otherwise there are certain coordinates which are 0 now this is not an operator this is what is called as natural inclusion again T is linear it is called a natural inclusion it is called a natural inclusion in particular this allows us to think of R m as sitting inside R n if n is greater than m you can think of R m as sitting inside R n so this is natural inclusion this is another example of a linear transformation. Let us look at other examples let us take one from the space of matrices let us say T from R m cross n the set of all the vector space real vector space of m by n matrices to the real vector space of n by m matrices defined by T of A equals A transpose T of A equal to A transpose this A transpose has been defined earlier if A is equal to A ij then A transpose is Aj i and so if A is m cross n then A transpose is n cross m this is linear that is because we need to verify T of x plus y equals T x plus T y let us look at T of A plus b T of A plus b by definition is the transpose of A plus b but the transpose can be verified to satisfy this formula A transpose plus b transpose ok this is a easy consequence of addition A transpose plus b transpose A transpose is T of A b transpose is T of b so this is additive T of A plus b is T of A plus T of b T of alpha times A is T of its alpha A transpose alpha A is multiplying alpha to each component of each term of the matrix A so that can be taken outside it is alpha A transpose remember it is a real case if it is a complex case you must take alpha bar outside alpha A transpose that is alpha T of A so T is linear we have verified so this one comes from transformation between vector spaces of matrices let us look at example from differential calculus let us take T from C prime 0 1 I usually say this is a complex valued space of complex valued continuous functions on the interval 0 1 C 0 1 C prime 0 1 space of complex valued continuous functions on 0 1 with the property that the first derivative is continuous I will consider this as a real space T from C 1 0 1 to C 0 1 again real space the mapping has T of a function f is f prime its derivative function T of f is f prime the derivative function T of sin theta is cos theta f comes from C 1 0 1 first of all this is well defined because T of f is f prime f if f is in C 1 0 1 does f prime belong to C 0 1 that is the case because of the definition of C 1 0 1 f prime is the first derivative that must be continuous so this f prime belongs to this so this is well defined this is a function to verify that this is linear comes from differential calculus D by D T of f plus G is D f by D T plus D G by D T D by D T of alpha times f is alpha D f by D T so this T is linear this is called the differential operator this T is called the differential operator okay so it is not just the superficial connection to differential calculus what we will see is that later when maybe in the next lecture we will discuss the notion of a range space of a linear transformation null space of a linear transformation there you will see that the null space of a linear transformation when T is a differential operator especially coming from constant coefficient that is it is a differential operator with constant coefficients then the null space is precisely the sets spanned by the solutions which are called the so called complementary functions of the differential equation okay so this connection is not just superficial okay this will be made clear later. So this is called the differential operator coming from differential calculus one from integral calculus and probably I will stop this list the last example is 10 let us say I have T from C 0 1 again for the sake of simplicity I will take this to be the real null space of continuous functions on 0 1 to R this time the domain vector space is one dimensional it is just R T defined by T of f is integral 0 to 1 f of T D T T of f is integral 0 to 1 f D T the Riemann integration now we know that this is well defined again because from integral calculus we know that every continuous function is Riemann integrable so the right hand side is well defined and you can verify easily that this is linear transformation that is integral for two functions f and g that are continuous integral 0 to 1 f of f plus f of T plus g of T D T is equal to integral 0 to 1 f T D T plus integral 0 to 1 g T D T T of that is T of f plus g equals T f plus T g T of alpha f is alpha times 0 to 1 f T D T that is alpha f alpha T of f rather so this is linear this is called an integral operator I will simply say integration integral transformation this is again a linear transformation okay probably one final example which sort of summarizes several of the previous examples not all of the I will state that as example 11 let me say T is from R n to R m is defined by so I have T of x equals A x where I am given an m cross n real matrix A I am given an m cross n real matrix A through this matrix I am defining a transformation this transformation is between R n and R m it is from R n to R m defined by this equation T x equal to A x this is matrix multiplication you see that if A is m cross n then A x is 1 1 cross m cross n m cross 1 okay that is a vector in R m so this is well defined most of the examples that we have discussed previously 0 transformation identity transformation the second example the third example reflection the fourth example rotation natural projection natural inclusion projection operator all these are particular cases of this for different choices of A this T is linear which follows by matrix multiplication this is linear okay this sort of summarizes all those examples now what is also true which is the most interesting part of linear algebra is that a certain converse is true that is if I have a linear transformation between finite dimensional vector spaces then there is a matrix which has a property that the that the transformation T satisfies this equation for that matrix okay if I have a linear transformation between finite dimensional vector spaces then there is a matrix A we can construct a matrix A such that this holds for the linear transformation T that we started with okay so let me just say that a certain converse is true and this is this holds for finite dimensional vector spaces okay so this list should probably convince you that linear transformations are indeed important objects before I proceed to the certain simple properties let me also consider this notion of what is linear sometimes is not really the linearity that we would like to have as illustrated here that is I want to give example 12 which is not really an example let us look at T from R to R defined by T of x equals x plus 1 okay T of x is x plus 1 the translation now we can plot this on R 2 that is you can call this as y then I have y equals x plus 1 now this is a straight line not passing through the origin okay you can verify that this T is not linear you can verify that this T is not linear in spite of the fact that intuitively in R 2 y equals x plus 1 is a line okay so if you have a formula representing a line in R 2 this does not necessarily correspond to a linear transformation this is just a simple point I wanted to illustrate okay in any transformation that transforms a line to a line is not necessarily a linear transformation is what I wanted to emphasize okay so this is not a linear transformation you can verify by simple examples that this T is not linear so anything that looks like linear is not necessarily linear on the other hand if it is straight line passing through the origin then this will be a linear transformation okay let us now look at some simple properties okay first property is the following so let me write down this theorem T from V to W T from V to W is linear then I have the following properties property 1 0 must be mapped to 0 this is the first property for instance you could use this property in that last example T of x equal to x plus 1 T of 0 is not 0 so that is not linear T of equal to 0 equals 0 property 2 we know that T of U plus V is T U plus T V this holds for U minus V also T of U minus V is T of U minus T of V and property 3 T of U plus V equals T U plus T V this can be extended to a finite sum T of alpha 1 U 1 plus alpha 2 U 2 etc let us say alpha k U k this is equal to alpha 1 T of U 1 plus alpha 2 T of U 2 plus etc alpha k T U k this additivity property that is a condition 1 that a linear transformation must satisfy can be extended to finitely many terms in fact linear combinations that is here these coefficients alpha k R in R U 1 U 2 etc U k they come from V let us quickly verify that these properties hold so I want to discuss the proof very quickly the look at the first part I will call U as T of 0 when U is T of 0 plus 0 T is linear so T of 0 plus 0 is T 0 plus T 0 I am calling this as U so I have U equals U plus U then from the first simple property of vector spaces it follows that U is 0 that is T of 0 is 0 that is the first property property 2 T of U minus V by definition this is T of U plus minus 1 times V minus V is minus 1 into V T is linear so T of X plus Y so that is T of U plus the constant is outside T of V minus 1 T of V that is happening in W so this is T U minus T V that is property 2 property 3 T of alpha 1 U 1 plus alpha 2 U 2 etc plus alpha k U k I will keep this as a vector maybe I will call it W then this is T of alpha 1 U 1 plus W I know that this is alpha 1 T of U 1 plus T of W then keep this as it is alpha 1 T of U 1 plus T of W formula for W T of alpha 2 U 2 plus etc plus alpha k U k I again have I will keep this as it is the rest of them I will call it W 1 and proceed alpha 1 T of U 1 plus alpha 2 T of U 2 plus T of W 1 where W 1 is alpha 3 U 3 plus etc plus alpha k U k proceed by induction etc this is alpha 1 T U 1 etc plus alpha k T U k okay so really simple property just making use of linearity definition of linearity okay a little more non-trivial properties of a linear transformation we will discuss next to motivate this property maybe I will give an example start with an example let us look at the functions sin x and cos x these are functions from R to R real valued functions of the real variable x these functions have the property that sin x equal to cos x at infinitely many points all those points starting from pi by 4 if you want and then you add 2 pi so there are infinitely many points x for which sin x equal to cos x okay for a linear transformation this kind of a thing will not be true for a linear if you have two so I have two really two functions sin x and cos x which coincide at infinitely many points but if you have two transformations T 1 and T 2 that coincide at all those basis elements then they must be the same linear transformation okay this is one important property which separates a linear transformation from a general function let me make this clear let V be a finite dimensional vector space and T 1, T 2 from V to W be linear transformations sometimes I will also call them maps these are functions so I have T 1 and T 2 linear maps from V into W no word about W V is finite dimensional suppose that I have a basis B let us say U 1 U 2 etc U n let this be a basis for V so see V is finite dimensional there is a basis considering of finite elements finitely many elements I am listing that basis suppose T 1 and T 2 satisfy the following equation if T 1 of U i equals T 2 of U i for all i 1 less than or equal to i less than or equal to n that is the transformations T 1 and T 2 coincide for the basis vectors the transformation T 1 and T 2 coincide for the basis vectors then we can show that T 1 is equal to T 2 then T 1 is equal to T 2 now contrast to this statement with the statement that I made to motivate this theorem sin x and cos x they are equal at infinitely many points but as functions they are not equal okay remember that T 1 is equal to T 2 means as functions these two are equal that is T 1 of x equals T 2 of x for all x and V as functions these two are equal they are one and the same one can also make the following informal statement from this theorem a linear transformation is completely determined by its action on any basis where I am assuming that the domain space is finite dimensional a linear transformation is completely determined by its action on any of its any of the basis of the domain space okay let us prove this quickly I want to say that T 1 is equal to T 2 so I will prove that T 1 of x equals T 2 of x for all x let x belong to V okay then I have a basis explicitly given script B so I can write x as alpha 1 U 1 plus alpha 2 U 2 plus etcetera alpha n U n okay let me now look at T 1 of x T 1 of this representation alpha 1 U 1 plus alpha 2 U 2 etcetera plus alpha n U n T 1 is linear so this is alpha 1 T 1 U 1 plus alpha 2 T 1 U 2 plus etcetera alpha n T 1 U n now I will make use of the fact that T 1 U 1 is equal to T 2 U 1 T 1 U 2 is equal to T 2 U 2 etcetera that is what is given T 1 and T 2 coincide for the basis vectors so this is alpha 1 T 2 U 1 plus alpha 2 T 2 U 2 plus etcetera plus alpha n T 2 U n again use of fact T 2 is linear to rewrite this as T 2 of alpha 1 U 1 plus alpha 2 U 2 plus etcetera plus alpha n U n but this is the x that we started with so this is T 2 of x so what we have shown is that T 1 of x is equal to T 2 of x for all x in V and so T 1 is equal to T 2 okay. Let me look at a numerical example to illustrate this result I want to give an example let us say T from R 3 to R 2 be such that T of the first basis vector is this T of the second basis vector I am in R 3 to R 2 so let us say minus 1 0 0 1 0 let us say this is 1 1 and T of 0 0 1 these 3 equations define T uniquely these 3 formulas define T uniquely what is the general formula for T of x I can write down because any x can be written as a linear combination of these okay so let us do that quickly let us take x in R 3 then x is I am following this notation consistently x 1 x 2 x 3 I can write this as x 1 into so okay see in our notation this is E 1 this is E 2 this is E 3 standard basis vector so this is x 1 E 1 plus x 2 E 2 plus x 3 E 3 any x is a linear combination of these the coefficients x 1 x 2 x 3 are given by the components of x I want T of x that is the question what is the general formula for T of x given x so T of x by definition is x 1 T of E 1 x 2 T of E 2 plus x 3 T of E 3 just plug in these values you get the formula for T of x so T of E 1 is minus 1 0 x 1 into minus 1 0 plus x 2 into 1 1 plus x 3 into 0 1 so you get a formula in terms of x this is minus x 1 plus x 2 second coordinate x 2 plus x 3 so this is T of x minus x 1 plus x 2 x 2 plus x 3 okay this is general formula if you know x you just plug in here you get T of x so the action of a linear transformation on a basis that is enough to determine the linear transformation completely okay let us also ask this question the answer will be given in the next this is the question I have let us say a basis V U 1 U 2 etc U n as before this is a basis of V I am given a set of vectors not necessarily a basis of W let us call them W 1 W 2 etc W n this is just a subset not necessarily a basis be a subset of W I have a basis for V and just a subset of W the question is does there exist a linear transformation a linear map T from V to W that takes the corresponding elements to the corresponding U I to the corresponding W I that is map such that T of U I is equals W I U I goes to W I does there exist a linear map T from V to V V to W such that this condition is satisfied okay for this to be satisfied do we need the conditions on W 1 etc W n okay if there exists a linear transformation is the transformation unique we will answer these questions in the next lecture.


One of the things I've been talking about this month, and it really came to me when I was on vacation, is I was reading a book by Ram Sharan called The Attacker's Advantage, and he dealt a lot with fear and uncertainty. And I always like to look up the definitions of words. And fear prohibits a person from taking action. Fear is an unpleasant emotion that is created by belief that someone or something is going to harm us in the future. Now, I focus in on a couple of those words. Feeling, emotion, belief. Those three things cause a person not to take action. I won't move because I got a fear that I'm going to be hurt in the future. I'm going to get my heart broken.  I have all these fears. Well, fear breeds uncertainty. And uncertainty is ambivalence, doubt, insecurity, vacillation, typically caused by an ignorance to something. And I don't mean that in a negative way. I just mean I'm ignorant to how it can help me. So when I have fear and uncertainty, I don't move. So as a professional, your job is to create certainty as much certainty as you can. And the way you create certainty is through mastery. The way you create mastery is through ongoing systematic practice, not cotton candy, but commitment. You get up and you do it over and over and over and you have a confidence about you that you know if they put themselves in your hands, you're going to take good care of them. You're going to get them to a much better state. Because of you at six months in the future, they're going to be much better off than they are today. That's what you're selling. And you can't give it to me if you don't have any. So I see a lot of people that are insecure. And this is the number one objection people have is I'm not moving forward because I don't have enough information and you haven't overcome my fear that something's going to hurt me in the future. Losing money has hurt me. Losing my reputation, losing my marriage, losing things I've worked for my whole life. Losing and fear stops me. So I vacillate. I go back and forth. Your job as a salesperson, your job as an entrepreneur is to overcome that fear in another person's mind through the certainty that you bring to the table. So when they vacillate, you need to let them know you got this because you're that good. You've been doing it for years. You know if they take action, they're going to be successful. That's how you overcome fear and uncertainty. Give them more information and give it to them any way they need it. Video, text message, Facebook, phone call, somebody else call them. Whatever you got to do, feed them more information. That's why they're not taking action because they don't believe that you can get them to the next level or they don't believe in their own self. They have not committed. It's all cotton candy versus commitment.


Arabs love their conspiracy theories. But often, these theories operate from a position of Arab victimhood and the result of external parties interfering with their progress, stability, and prosperity. Whether it's Western colonialist intervention, Islamic anti-Semitism, petroleum induced, or war on terrorism, every instant of the Arab world's desperate holding onto conspiracies is a means to justify their current and mismanaged predicaments. But when it comes to conspiracies internal to the Arab world, where it's an Arab-on-Arab concept, Hamush, nothing, silence, Arabs apparently don't conspire against themselves. And at this point is where we arrive at our story, to share a tale that had been hidden from Arabs for 40 years, and up until 1986, when those involved stepped forward and started to speak, and the more that spoke, the more facts were corroborated, and eventually this theory could no longer be ignored. This is the real story of how in 1935, the Red Book Collective, a secret movement, set the foundations for real and effective Arab nationalism over the next 40 years. In the late 19th century and beginnings of the 20th century, the concept of the motherland saw resurgence in Arabs. Having seen the various extraordinary advancements in Europe during the 1800s, as well as the static reality of the Ottoman Empire, intellectual movements started taking shape, criticizing the state of Arab identity and pride, looking for an opportunity to return the motherland to the forefront of the global stage. Conditions worsened after World War I, as the region was divvied up amongst the victors into mandates. Ignoring the promises made to the Arabs who had just assisted them in revolting against the Ottomans, the Imperial British and French carried on with their colonial aspirations by territorially partitioning the region. A single nation for the Arabs was not meant to be. Further Arab strife came about when the aggressive actions of the British and the French further alienated the populations, leading to anti-colonial engagements and uprisings in Palestine, Lebanon, Syria, Egypt and Iraq. These circumstances led to the formation of several organizations that reflected the sentiments of a growing nationalism. It's the concept of a single unified Arab nation that prioritizes Arab culture, civilization and language. It calls for the reinvention of Arab society in line with modern interpretations of Islam and implements a new religious rationality more in line with the times. From the Atlantic coast of North Africa to the Indian Ocean, all Arabs were to be bound by a common identity of ethnicity, language, religion, culture, history, geography and ultimately politics. A priority was to overcome underdeveloped education, healthcare and infrastructure through major improvement of all civil services. As the idea came about during the Western colonial occupation, Arab nationalism commanded the elimination of all foreign influence that had degraded the Arab way of life, ethics, values, dignity and unity. All endeavors by any and all Arabs would be to achieve an independence granting self-rule, governance and fulfillment of the cause. Please subscribe to our channel as it would support us greatly in generating more content that documents our Arabian and Muslim heritage, history and culture. Now back to our story. In the beginning of 1935, within the mandate of Syria and Lebanon and on the campus of the American University of Beirut, a meeting was taking place in secrecy and in attendance was Konstantin Zurek, a professor of history, with Khadim and Tafiyuddin Saleh, both members of the Lebanese parliament. The purpose was the establishment of an Arab nationalist movement that would infiltrate the various existing political structures around the Middle East to instill a certain fundamentalist belief on the concepts of Arab nationalism. Six months later, after the tight-lipped recruitment of educated men and professionals by the three founders, the larger group of ten reconvened. But now, with a printed charter book, with a red cover, at the ready, documenting their mission statement and modus operandi, and hence their label, the red book collective. The five precepts that the collective's charter included were, The Arabs are one nation, those whose mother tongue is Arabic and those who reside in the Arab lands and have no objections to join the Arab nation. Arab lands are for Arabs alone. Arab lands are those whose populations speak the Arabic language, be them in Asia or Africa. To accept the choice that all Arab regions shall be unified into a federation structure. The intended Arab nation is to be a civil state and not a religious one, and shall be governed in such a manner. The universal freedoms such as thoughts, speech, writing and gatherings, as well as ownership and others, are protected by the rule of law for the benefits of the individual and society. In addition, there were three ancillary goals, and they are, liberating the Arab lands from foreign control, liberating the Arab society from ignorance, poverty, illness and other depravities, spreading the concept of Arab nationalism. On the surface, the collective's mission was the same as that of other Arab nationalist movements, except theirs identified two major differences. The first was to deal in total secrecy, to realize their agenda and its radical nature, thus avoiding the limelight. And second, what wouldn't succeed in diplomacy would have to be resolved by force. Over the next two years, the efforts to expand started within the American University itself, and targeted the various powerful student organizations saturated with highly politicized and influential Arab graduates from across the region. This strategy of infiltrating existing organizations, be them educational, social or political, became a very successful strategy in an accelerated expansion around the region over the next two years. Between 1936 and 1937, branches for the collective were opened in Syria, Palestine, Iraq and Kuwait. These entities operated normally, but also as fronts to recruit and embed the necessary network of cells and agents that the collective needed to become the king-making secret society that it intended on being from the outset. With aggressive and widespread expansion came problems for the collective. Its base of operations shifted to Iraq and saw a new leadership that miscalculated its historically successful method of management, and soon infighting began. In 1945, after a mere decade of existence, the collective collapsed and ceased to exist as an organization. The doctrine in the Red Book, though, was still out there, within the network of secret operatives that were still in place and would continue operating within the same mission, the rise of Arab nationalism, embedded within the various political societies of Arab nations. From the early 1950s, country after country became independent through military coups that led with announcements of Arab nationalism and unity. Was this a coincidence? After a mere decade of the collective's establishment and its intense infiltrations across the region, all these coups, ending monarchies, to then call for the unification of the Arab lands and people? It would be a misrepresentation to say that the collective was the sole voice for Arab unification. There were definitely Arab nationalist thinkers, missions and organizations outside of the Red Book collective. But what can't be denied is how effective this collective was in infiltrating the most highly connected, educated, and powerful of the Arab peoples and institutions. The various members of the collective most became extremely high-racking figures in their respective countries, from members of parliament, ambassadors, to ministers, prime ministers, and even in one case, president of a nation. Due to the collective secrecy, we will never know the absolute truth about the extent of their network or recruitment. But what we do know is that the timing of the establishment of the Red Book collective and the rise of the Arab nationalists a decade later could not be such a coincidence, and maybe not such a conspiracy after all.


Everybody forced here have been twenty one twenty two select basketball hobby hybrid ten bucks number 18 Check out these results anytime of late sports cards.com. Let's get started This gets you it's probably not gonna recap this one unless we had a bunch of monsters just behind and this is gonna be a pretty quick Break here. So if you're watching this time of bones wise camp Quartz side disco Tramond green Concourse rookie disco Jaden springer sixers. There's Davion First color coming up. There's a rookie concourse disco Kaminga for the Warriors and a blue Donovan Mitchell jazz for two of 25 nice Donovan Jake with a chair again over here. Sorry Concourse disco sexton Cavaliers silver concourse yoke itch nuggets It's like looks like him bead might might take MVP this year. Trey Murphy the third. He's he's been ballin recently red Kevin Porter jr. Quartz side Rockets 30 of 49 Scotty Barnes concourse concourse rookie disco Quentin Grimes nicks and a premier level silver of Eric Bledsoe No, there's still some space box Cameron Thomas Concourse disco all the depot with a silver concourse rookie Franz Wagner Franz Have you seen that in the magic? Man, I'll show you Mobley concourse rookie Cleveland's Premier level rookie disco Alperin shengun. There you go Rockets And Amy Lillard, I'm sure you would know exactly what I meant if you had seen it Franz So good doing pretty well man, how are you? Nice Saturday concourse red on a dimensional 26 at 49. Okay jazz. Okay Okay Donovan in the next oh, we all thought was gonna happen and then they just didn't want to trade that much. It was weird Jalen Suggs Concourse concourse disco Dallin of Banton Raptors rookie and a key on Johnson rookie blue disco Clippers color match 22 of 25 Come on FAU. Let's go FAU Concourse disco Rondo Cavs and a silver concourse rookies. I are Williams Grizzlies. Oh Boy See, I haven't I haven't nothing happened to me yet. I I hope that that stays the same But yeah, you gotta you gotta spill the you gotta spill the tea now Shade a day run sharp, but not shade on sharp. That's different draft last rookie premier level disco for the Nets Let's do Concourse rookie disco Tram and for the Thunder I was on Twitter earlier and I saw like a The coyotes are moving to to Houston full-time starting next season Yeah Conqueror circuit disco bones island for the Nuggets silver premier level the mellow ball And I I was like what and I was like, oh, it's April 1st It's like they wouldn't announce that in the mid-season, you know would go to any games key concourse disco Garza Pistons Conqueror silver rookie Garuba Rockets. I saw somewhere Thomas Cameron Thomas rookie premier level disco for the Nets with Jeter People were saying is this an April Fool's joke? He did what they replace Frank Thomas with their jeter on like the fox like yeah, like It's Thomas is done like it no more. Yeah, they're swapping him in jeter Like Frank Thomas premier level disco Clint Capela for the Hawks I don't think Jeter has enough like personality for that. That's what people were saying like it unless Cuz like Poppy has the personality for stuff like that concourse disco Jimmy Butler for the heat He's like the entertaining one on that a rod a rod is very like Yeah, a rod's very very like vanilla now like he you know Silver terror was here field level for the Hornets blue rookie premier level friends 2325 for the magic Big card Orlando going to Eric W. Franz I Know Kyle's watching he's laughing Big-time frowns 23 one off of the Jersey number still a big card concourse red Davion Mitchell rookie Kings 27 to 49 and a silver concourse look at Donchich Mavericks. I Agree Vaughn I agree. I don't dislike Jeter. I just him and a rod are very like Vanilla like I don't know they That's not a negative thing that I just don't know if the style is gonna work out together like hurt hurt a rod and Poppy were Really good together. I think a court side disco Deandre Hunter for the Hawks Sab knows what I'm talking about the fronds thing sabnos from Concourse disco Steph Curry Warriors Shen goon Concourse disco book night with a green behind it's a green after the book night. That's gonna go this way Rookie No No rookie Very nice Michael Porter Jr. Five of five Denver Green court side.  Let's try one FAU Concourse disco rookie Josh Christopher Rockets Concourse disco Mark and in Cavaliers premiere level rookie silver day run sharp Brooklyn Franz concourse giddy court side red concourse rookie xire Williams Grizzly's 24 49 and a silver Chris Duarte rookie Pacers concourse Shengun concourse with a premier level rookie disco Evan Mobley for the Cavs Thomas three Premier level rookie disco. I ought to soon move bulls Kaminga Concourse disco Seth Curry the Marta Rosen Seth Curry for the Sixers there concourse disco K Love Cavs Premier level silver. Nikhil Alexander Walker pelicans Alright, so five to go on corkings everybody again. Just reduce that price earlier two teams of spot Breaking after prism which is after this dortier chamber concourse disco the Thunder silver Caruso a Reeves rookie court side disco Lakers concourse over Duarte again Pacers Concourse disco Jordan Clarkson for the Jazz concourse over Seth Curry Sixers Coming concourse disco Corey Kispert wizards premier level silver Trey Young Hawks now kabooms and basketball are in Crown Royale concourse disco rookie Clinton Grimes nicks concourse over Rondo Cavs Book Knight premier level disco Hornets bones in there Primo concourse rookie disco for San Antonio Yeah, there's some other short prints. They're just not like Monster chases. They're just like I forget the name of them concourse disco Carlin three towns T Wolves premier level silver Blake Griffin It's not fresh paint fresh paint is the auto Kaminga But he healed concourse disco Kings and a silver Kai Jones Hornets show off two cards here There's some nice color. These are the these are by far the nicest ones though Nuggets and magic congrats got green court side Michael Porto, Jr. for the nuggets and a blue premier level Franz Wagner magic 25 23 25 Franz and five of five MPJ. Thanks again everybody on a prism


Get them their dream gifts. Same-day home delivery, seven days a week with Argos. You're good to go.