Transformations and matrix multiplication
Distributive property of matrix products
Let's say we have three matrices, A, B, and C. And let's say that B and C are both m by n matrices, and that A is a, let's call it a k by m matrix. And what I want to do is figure out whether matrix products exhibit the distributive property. So let's test out A times B plus C. And of course these are all matrices. So B, just to make things clear, the matrix B could be represented as just a bunch of column vectors, B1, B2, all the way to Bn. And the matrix C can also be represented as just a bunch of column vectors. So could the matrix A, but I don't have to draw that just yet. So the matrix C could just be represented as a bunch of column vectors, C1, C2, all the way to Cn. Maybe I should've drawn this taller. These are column vectors, so they actually have some verticality to them. I think you've seen that multiple times. So what is A times B plus C? Well, let's figure out what B plus C is. This is equal to A times B plus C. When you add B plus C, the definition of matrix addition is, you just add the corresponding columns. Which essentially boils down to adding the corresponding entries. So the first column is going to be equal to B1 plus C1. The second column is going to be B2 plus C2. And you're going to go all the way to the nth column. It's going to be Bn plus Cn. Now by our definition of matrix-matrix products, this product right here is going to be equal to the matrix, where we take the matrix A and multiply it by each of the column vectors of this matrix here, of B plus C. Which as you can imagine, these are both m by n. In fact they both have to have the same dimensions for this addition to be well defined. So this is going to be an m by n matrix. I already you told you that A is a k by m. And we know this is well defined because A has the same number of columns as B plus C has of rows. So this is well defined. And this is going to be equal to-- let me switch colors again-- A times the column vector B1 plus C1. The second column is going to be A times the column vector B2 plus C2. I'm running out of space. This is going to be all the way to A times the column vector Bn plus Cn. This is our definition of matrix-matrix products. You just take the first matrix and you multiply it times each of the column vectors of the second matrix. And we can say that because we've already defined matrix-vector products. So what is this thing on the right equal to? I'll keep switching colors. We know that matrix-vector products exhibit the distributive property. I don't even remember when I did that video. But we've assumed it for a while. It's a very trivial thing to prove. So each of these columns is going to be equal to, let me write this way. This guy right here can be re-written. The first column is going to be A times column vector B1, plus A times the column vector C1. This term right there is the same thing as that term right there. The next one is going to be AB2 plus matrix A times the vector C2. And then the nth column is going to be the matrix-- keep going-- A times the column vector Bn, plus matrix A times the column vector C. Just like that. Now we can write this matrix as the sum of two different matrices. So what is this going to be equal to? This is equal to-- let me see, I'll just write it right here-- AB1 as the first column, AB2 as the second column, all the way to ABn as the third column. So that's these terms right there. And then if I were to add to that the matrix A times vector C1, A times the column vector C2-- these are just the different columns of this matrix-- and we just then have the matrix A times the column vector Cn. These represent these terms. So clearly if I add these two matrices, I just add the corresponding column vectors and I'll get this matrix up here. But what is this equal to? This right here, by definition, this is the matrix A times the matrix B. The definition of matrix products is you take the first matrix and multiply times the column vectors of the second matrix. And by the same argument, I guess you could say, this is equivalent to A times C. And all of this-- remember we just had a bunch of equal signs-- is equal to A times B plus C. So now we can say definitively that as long as the products are well defined and the sums are well difined, so they all have to have the correct dimensions, that A times B plus C is equal to AB plus AC. So matrix products do exhibit the distributive property, at least in this direction. And I say that because remember matrix products are not commutative. So we don't know necessarily that B plus C times A is equivalent to that. In fact, most of the time these two things are not equivalent. So we don't know quite yet that if we reversed this, whether it's still going to exhibit the distributive property. So let's try to do that. And I'll do it a little bit quicker, because I think you know the general argument here. So let's take B plus C times A. And I'll just write A as its column vectors. A1, A2, all the way to-- A has m columns if I remember correctly, right A has m columns-- so all the way to Am. And by the definition of matrix products, this is going to be equal to the matrix-- B plus C is just a matrix, right? We can represent it as the sum of two matrix, but it is just a matrix. So it's B plus C times each of the column vectors of A. So it's going to be equal to B plus C times A1, B plus C times A2, all the way to B plus C times An. And, once again, it was many videos ago that I think we showed that matrix-vector products are distributive, so we can just distribute this vector along these two matrices. And if I haven't proven it yet, it's actually a very straightforward prove to do. So we could say that this is equal to BA1 plus CA2. That's the first column. The second column is B times A2, plus C times A2, all the way to B times An, plus C times An. And then what is this equal to? I'll write it out. This is equivalent to B times-- this is A1, no this is an A1 right here-- A1, and then B times A2, all the way to BAn plus the vector C times A1, C times A2, all the way to C times An, right? This guy represents these terms right there, and this guy represents the first terms in each of these column vectors. And this, by the definition of matrix products, is just equivalent to BA, and then this is just equivalent to CA. So now we've seen that the distributive property works both ways with matrix-vector products. That B plus C times A is equal to BA CA, and that A times B plus C is equal to AB plus AC. Now the one thing that you have to be careful of is that these two things are not equivalent [UNINTELLIGIBLE]. We just figured out that this guy is equal to BA plus CA. So the distribution works in both ways. But when you're dealing with matrices, it's very important to keep your order. So this is going to be you have the A second here. So it is BA plus CA. You can't say that this is equal to AB plus AC. You can't just switch these up. Because we've shown multiple times, or we've talked about it multiple times, that matrix products are not commutative. You can't just switch the order of the products. But we've at least shown in this video that the distributive property works both ways.