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Video transcript
So we're multiplying matrix A here by vector w. And they ask us what is A times w? So let me get my scratch pad out. And let's think about this. So we have matrix A times vector w. So just as a reminder, Aw-- and I'll write it in bold right over here-- Aw. And I could write it either as a vector like that, or I could bold that as well. Aw. It's lowercase for a vector, uppercase for a matrix. This is the same thing as matrix A times vector w-- let me do that same color-- which is the same thing. This is matrix A. So it's, A is 0, 3, 5, 5, 5, 2 times vector w. W is the vector 3, 4, 3. So we have a matrix with two rows and three columns, so it is a 2 by 3 matrix, times a vector. And this is a column vector. It is three rows in one column. So you could view this as a 3 by 1 matrix. And so, deciding whether this is even a valid operation, it's just like multiplying two matrices. You can view this column vector is really just a 3 by 1 matrix. And the only way that matrix multiplication is defined is if the columns in this matrix is equal to the rows in this matrix. And we see that is the case, this vector. This matrix, matrix A, has three columns: 1, 2, 3. And this and vector w has exactly three rows. So matrix multiplication, or matrix times vector multiplication, is defined here. And the way that we're going to do it, we're going to end up with a 2 by 1. You could call it a 2 by 1 matrix. It's going to have two rows in one column. Or you could view it as a column vector, when you have two rows and one column. But either way, let's actually think about how we would compute it. So it's going to have two rows and one column. Then I'm going give a lot of space so that we can actually do the calculation. So the top entry right over here, we're going to get the row information from our first matrix, and then the column information. There's only one column here. But we're going to get that from this matrix or this vector, whatever you want to call it. So we're going to have 0 times 3, plus 3 times 4, plus 5 times 3. Now the second entry right over here is going to be the second row here. Essentially, the dot product of that with this vector. And if you don't know what the dot product is, I'm essentially about to do that. It's going to be-- you take each corresponding term, take their product, and then add up everything together. So you have 5 times 3, plus 5 times 4, plus 2 times 3. And this is going to simplify to-- This top term, this is a 0. This is 12 plus 15, which is 27. And then the second term, this is 15 plus 20 plus 6. So this is 35 plus 6 is 41. So it's a column vector, two rows, 27 and 41. Now let's input that, 27 and 41. So we get 27. And here we can put 41, and check our answer, and we got it right.