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Sums and scalar multiples of linear transformations

Video transcript
Let's say I have two transformations. I have the transformation S, which is a function or a transformation from Rn to Rm, and I also have the transformation T, which is also a transformation from Rn to Rm. I'm going to define right now what it means to add the two transformations. So this is a definition. Let me write it as a definition. I'm going to define the addition of our two transformations. So if I add our two transformations, the addition of two transformations operating on some vector x, this is a definition. I'm going to say this is the same thing as the first transformation operating on the vector x plus the second transformation operating on the vector x. And obviously, this is going to end up being a vector in Rm, so this whole thing is going to be a vector in Rm. By definition, this S plus T transformation is still a transformation because it takes an input from Rn. It's still a transformation from Rn to Rm. Now let me make another definition. Let me define -- I'll do it in green. Maybe I'll do it in purple. I'm going to define a scalar multiple of a transformation. So I'm going to define, let's say c, where c is just any real number. c times the transformation S of some vector x, I'm going to say that this is equal to c times the transformation of x. And so similarly, the transformation of x obviously is going to be in Rm. So if you multiply any vector in Rm times some scalar, you're still going to have another vector in Rm. So luckily for us, this definition of a scalar multiple-- so if I have this new transformation called c times S, this is still a mapping from Rn to Rm. This is still a vector in Rm and this is still a vector in Rn. Fair enough. Now, let's see what happens if we look at their corresponding matrices for these transformation. We've seen in a previous video that any linear transformation can be represented as a matrix vector product. So let's say that S of a vector x is equivalent to the matrix A times that vector x. And let's say that T of x is equal to the matrix B times the vector x. And, of course, since both of these guys are mappings from Rn to Rm, both of these matrices are going to be m by n. Both of these are m by n matrices. Now, let's just go back to these definitions that I just constructed. What is S of T of x? That can then be written as-- so let me write it this way. I'll do it in that same color. So you have S-- I was going to do it in red. Maybe I'll do it right here. You have S plus T-- that's a capital T. S plus T of x-- I'm just re-writing this up here -- is equal to S of x plus T of x, or the transformation T of x, which we now know is equal to these two things. So this is equal to this term right there. The transformation S of x is equal to Ax. That's that one right there. And then the transformation T of x is equal to B, the matrix B times x. Now, what are these things? Let me write our two matrices in a form that you're probably familiar with right now. Let's say the matrix A is just a bunch of column vectors: a1, a2, all the way to an. And similarly, the matrix B is just a bunch of column vectors. The matrix B is b1, b2, all the way to bn. These are each column vectors with m components, one for each of the rows, and there's n of these because there are n columns in each of these vectors. So when you multiply this guy times-- let me make it very clear. If I multiply an x, the vector x is going to look like this. The vector x is going to be x1, x2, all the way down to xn. And we've shown this multiple, multiple times. It's a very handy way of thinking about matrix vector products. But we know that this product right here can be written be as each of these scalar terms in x times its corresponding column vector in A. I've done this, and it's probably the fifth video that I'm doing this. So this can be written as x1, x1 times a1 plus x2 times a2, all the way to xn times an is equal to this. That's what ax can be rewritten as, as kind of a weighted combination of these column vectors where the weights are each of the values of our vector x. And I have to add this guy to bx. So bx, by the same argument, so plus is just going to be-- let me do it in the blue. It's going to be x1 times b1 plus x2 times b2, all the way to xn times bn. Now, what is this equal to? Well, we know that scalar multiplication times vector exhibits the distributive property, so we can just add these two guys right here and factor out the x1. And what do we get? We get this is equal to-- this whole expression right here, let me draw a line here, because I'm not saying this matrix is equal to that. I'm saying that this is equal to this, is equal to this term plus this term, which is equal to x1 times a1 plus b1, plus x2 times a2-- I'm just adding these two terms up-- x2 times a2 plus b2, all the way to plus xn times an plus bn. So what is this thing equal to? Well, this is equal to some new matrix, and let's define this new matrix. This is equal to some new matrix-- I'll make it pretty big right here-- times our vector x . I'll do the vector x in green. Vector x we know is x1, x2, all the way down to xn. But what is the new matrix going to be? Well, this product is going to be each of these scalar terms times the column vectors of this matrix. So these guys right here are the columns of my matrix. This thing is equivalent to a matrix where the first column right here is a1 plus b1. We're essentially adding the column vectors of those two guys. The second column right here-- let me draw a little line right there to show you that these are different expressions. The second one would be a2 plus b2, and then we'll just have a bunch of them, and then the last one will just be an plus bn. So what happens is that, by definition, when I added these two transformations, I just used their corresponding matrices. And I said you know what? The addition of these two transformations created a new transformation that is essentially some matrix times my vector, and that matrix ended up being the sum of the corresponding column vectors of our two original transformation matrices, right? This new matrix that I got, and I haven't defined matrix addition yet, but we got here just by thinking about vector addition. This matrix is constructed by adding the corresponding vectors of the matrices A and B. Now, why did I go through all of this trouble? Well, I can make a new definition here that'll make everything fit together well. I'm going to define this matrix right here as A plus B. So my new matrix definition, if I have two matrices that have the same dimensions, and they have to have the same dimensions, I'm defining A plus B to be equal to some new matrix where you add up their corresponding columns. So a1 plus b1, just like what I did here, I don't have to rewrite it, all the way up to an plus bn is the last column. And you've seen this before in your algebra II class, but I wanted here to do it, because this shows you the motivation for it. Because now we can say that the sum of two transformations, So S plus T of x, which is equal to S of x-- this is a vector-- S of x plus T of x, which we know is equal to A times x plus B times x, we can now say is equal to, because it's equal to some new matrix, which we can now call A plus B times x, right? I just showed this part is from the definition of our transformations into some of our transformation that I defined earlier in this video. And then when we just worked this out and kind of expressed these products as products of or as weighted combinations of the column vectors of these guys, we got to this new matrix. And I defined this new matrix as A plus B. And I did that because it has this neat property now because now the sum of two linear transformations operating on x is equivalent to, when you think of it is a matrix vector product, as the sum of their two matrices. Now, let's do the same thing with scalar multiplication. We know that c times our transformation of x by definition I'm saying is c times the transformation of x. So c times whatever vector this is in Rm. And so we know that S of x can be rewritten as Ax, so this is c times A times x. And we know that Ax can be rewritten as this is equal to c times x1 times the first column vector in a, so a1 plus x2 times a2,xn all the way to plus xn times an. Now, what is this? This is just scalar multiplication. We can just distribute this c. and then what do we get? We get x, and multiplication is associative. c is a scalar, x1 is a scalar, so we can switch them around if we want. We know that scalar multiplication is distributive, so we can write this as x1 times ca1 plus x2 times ca2, all the way to xn times can. Now, what is this equal to? This is equal to some new matrix times x. This is equal to some new matrix-- let me make that here-- times x1, x2, all the way to xn. And what is that new matrix? What are the columns of the new matrix? Well, the columns are now that, that, all the way to that. So the columns of this new matrix are ca1, ca2, all the way to can. Now, why would I go through this exercise? Well, wouldn't it be nice, I already said that by definition a scalar multiple of a transformation is equal to the scalar times a transformation of any vector that you input into it. And, of course, that is equal to c times Ax. Now, wouldn't it be nice if I could define this thing as some new matrix times a vector x, right? Because this should also be a linear transformation. And this new matrix I'm going to define. This is a definition again. I'm going to define this new matrix as being c times A. So now we have this definition that c times A, if I take any scalar times any matrix A, it's just equal to c times each of the column vectors. And we know what happens when you take a scalar times each of the-- just let me write this. This is equal to c times a1, c times a2-- I'm just rewriting what I just wrote there-- all the way to c times an. But what is this in effect? We know that when you multiply c times a vector, you multiply the scalar times each of the vector's elements. So this is the equivalent of multiplying c times every entry up in this matrix right here. And with this video, you know, you're probably saying, hey, Sal, I already knew how to-- in algebra II in tenth grade or ninth grade, I already was exposed to multiplying a scalar times a matrix or adding two matrices with the same dimensions. Why did you go through all of this trouble of the defining the sum of transformations and the sum of matrices? And I went through the trouble because I wanted you to understand that there's nothing-- I mean, it is natural, but there's nothing about the universe that said matrices had to be defined this way. Matrix addition, or matrix scalar multiplication, or the addition of two transformations. I wanted you to see the mathematical world has constructed it in this way because it seems to have nice properties that are useful. And that's what I've done in this video. In the next video, I'll do a couple of scalar multiplications and matrix additions just to make sure that you remember what you had learned in your ninth or tenth grade algebra class, but you'll find that the actual operations are almost trivially simple.