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# Linear transformations

## Video transcript

you now know what a transformation is so let's introduce a more of a special kind of transformation called a linear linear transformation transformation it only makes sense that we have something called a linear transformation because we're studying linear algebra we already had linear combination so we might as well have a linear transformation and a linear transformation by definition is a transformation which we know is just a function we could say it's from the set RM let me say it from RN to RM and it might be obvious in the next video why I'm being a little bit particular about that although they are just arbitrary letters where the following two things have to be true so if some if something is a linear transformation something is a linear transformation if and only if the following thing is true let's say that we have two vectors say vector a and let's say vector B are both members of RN so they're both in our domain so that this is a linear transformation if and only if if I take the transformation of the sum of our two vectors if I take if I add them up first that's equivalent to taking the transformation of each of the vectors taking the transformation of each of the vectors and then summing them that's my first condition for for this to be a linear transformation and then the second one is is that if I take the transformation of any scaled up version of a vector so let me just multiply vector a times some scalar some real number see if this is a linear transformation then this should be equal to C times the transformation of a that seems pretty straightforward let's see if we can apply these rules to figure out if some actual transformations are linear or not so let me define a transformation let's say that I have the transformation T part of my definition I'm going to tell you it's it maps from R to 2 let's do something let's just say it maps from R to - our - and my deaf and it Maps so if you give it a 2-tuple right it's domain is 2-tuple so you give it an X 1 and then X 2 let's say it Maps 2 so this will be equal to or it's associated with let me it's associated with let me just say let's say X 1 plus X 2 and then let's just say it's 3 times X 1 is the second tuple we could have written this more in vector form this is kind of our tuple form we could have written it and it's good to see all the different notations that you might encounter you could write it a transformation of some vector X where the vector looks like this X 1 X 2 let me put a bracket there it equals it equals some new vector X 1 plus X 2 and then the second component of the new vector would be 3 X 1 that's a completely legitimate way to express our transformation and the third way which I never see but to me it kind of captures the essence of what a transformation is it's just a mapping or it's just a function we could say that the transformation is a mapping from any vector in r2 that looks like this X 1 X 2 2 and I'll do this notation to a vector that looks like this X 1 plus X 2 and then 3 X 1 all of these statements are equivalent but our whole point of writing this is to figure out whether T is linearly independent so I want to sorry not linearly independent whether it's a linear transformation I was so obsessed with linear independence for so many videos it's hard to get it out of my brain in this one whether it's a linear transformation so let's test our two conditions I have them up here so let's take T of let's say I have two vectors a and B they're members of r2 so let me write it a is equal to a 1 know a 1 a 2 and B is equal to B 1 be - sorry that's not a vector I have to make sure that those are scalars these are the components of a vector and B - so what is a 1 + B I'm sorry what is a vector A plus vector B brains malfunctioning all right well you just add up their components this is the definition of vector addition so it's a 1 plus B 1 out of the first components and the second components is just the sum of each of the vector second components a2 + b2 nothing new here but what is the transformation of this vector so the transformation the transformation of vector A plus vector B we could write it like this that would be the same thing as the transformation of this vector which is just a 1 plus B 1 + a2 + b2 which we know it equals a vector it equals this vector where what we do is for the first component here we add up the two components on this side so the first component here is going to be these two guys added up so it's a 1 plus a 2 plus B 1 plus B 2 and then the second component by our transformation or our function definition is just 3 times the first component in the NR in our domain I guess we could say it so it's 3 times the first one so that's going to be 3 times this first guy so it's 3 a 1 plus 3 B 1 fair enough now what is the transformation individually of a and B so the transformation of a is equal to the transformation of a let me write it this way it's equal to the transformation of a 1 a 2 in brackets that's our another way of writing vector a and what is that equal to that's our definition of our transformation right up here so this is going to be equal to the vector a 1 plus a 2 and then 3 times a 1 just come straight out of definition I essentially just replaced an X with a z' by the same I guess argument by the same argument what is the transformation of our vector B well else this is going to be the same thing with the a is replaced by the B so it's the transformation of our vector B is going to be B you know B is just B 1 B 2 B 1 B 2 so it's going to be B 1 plus B 2 and then the second component and the transformation will be 3 times B 1 now what is what is the transformation of vector a plus the transformation of vector B what's this vector plus that vector and what is that equal to well this is just pure vector addition so we just add up their components so it's a 1 plus a 2 plus B 1 plus B 2 that's just that component plus that component the second component is 3 a 1 we're going to add it to that second component so it's 3 a 1 plus 3 B 1 now we just showed you that if I take the transformation separately of each of the vectors and then add them up I get the exact same thing as if I took the vectors and added them up first and then took the transformation so we've met our first criteria that the transformation of the sum of the vectors is the same thing as the psalm of the transformations now let's see if this works with the width with a random scalar so we know what the transformation of a looks like what is the transformation but what does C a look like first of all I guess that's a good place to start C times our vector a is going to be equal to C times a 1 and then C times a 2 that's our definition of scalar multiplication times a vector so what's our transformation let me go to a new color what is our let me do a color I haven't used in a long time white what is our transformation of see a going to be well that's the same thing as our transformation of ca1 ca2 which is equal to a new vector we're the first term let's go to our definition is you sum the first you sum the first and second components and the second term is 3 times the fur component so our first term you sub them so it's going to be C a1 plus C a2 and then our second term is 3 times our first term so it's 3 C a1 now what is this equal to this is the same thing we can kind of you could view it as factoring out the C this is the same thing as C times the vector a1 plus a2 and then the second component is 3 a1 but this thing right here we already saw this is the same thing as the transformation of a transformation of a so just like that you see that the transformation of C times our vector a for any vector a and r2 this is anything in r2 can be represented this way it's the same thing as C times the transformation of a so we've met our second condition that it doesn't that when you when you well I just stated it so I don't have to restate it so we meet both conditions we meet both conditions which tells us that this is a linear transformation and you might be thinking ok Sal fair enough that was you know how do I know that all transformations aren't linear transformation show me something that that won't work and here I'll do a very simple example let me define my transformation and just to make it oh let me let me do one let me make it let me define a transformation I'll do it from r2 to r2 just for just to kind of compare the two I could have done it from R to R if I wanted a simpler example but I'm going to define my transformation let's say my transformation of the vector x1 x2 let's say it is equal to let me just say x1 the squared and then 0 just like that let me see if this is a linear transformation so the first question is if if let's let's take what's my transformation of a vector a so my transformation of a vector a where a is just the same a that I did before it would look like this it would look like a 1 squared and then zero now what would be my transformation would be my transformation if I took C times a C times a well this is the same thing as C times a 1 and C times a 2 and by our transformation definition if I take I sorry the transformation of C times of this thing right here because I'm doing the transformation on both sides and by our transformation definition this will be just be equal to a new vector that would be in our codomain where we the first term is just the first term of our input squared so it's C a1 squared C a1 squared and the second term is 0 and what is this equal to let me switch colors this is equal to C squared a1 squared and this is equal to 0 now let's let's see if we can factor out if we well if we can assume that C does not equal 0 this would be equal to what this would be actually it doesn't even matter we don't even have to make that assumption so let's this is the same thing this is equal to C squared times the vector a 1 squared 0 which is equal to what this is equal this expression right here is the transformation of a so this is equal to C squared times the transformation of a let me do it in the same color times the transformation of a so what I've just showed you is if I take the transformation of a vector being multiplied by a scalar quantity first that that's equal to for this T for this for this transformation that I've defined right here that's equal to C squared times the transformation of a and clearly this this this statement right here for this choice of transformation conflicts conflicts with this requirement for a linear transformation have a see here I should see a C here but in our case I have a C here and I have a C squared here so clearly this negates that statement so we're definitely not dealing so this is this is not a linear transformation and just to get a gut feel if you're just looking at something whether it's going to be a linear transformation or not if the transformation just involves linear combinations of the different components of the inputs you're probably dealing with the linear transformation if something if you start seeing things where the components start getting multiplied by each other or you start seeing squares or exponents you're probably not dealing with the linear transformation and then there are some functions that might be in a bit of a gray area but it tends to be just linear combinations are going to deal lead to a linear transformation but hopefully that gives you a good sense of things and this leads up to what I think is one of the neatest outcomes in the next video