Which of the following statements about matrix-vector...

Which of the following statements about matrix-vector products is FALSE? (Hint: If you are not sure, try some examples.) The product Av of an mxn matrix A with an nx1 column vector v is a linear combination of the columns of A. The product u^T v , where u and v are nx1 column vectors is defined and equal to a 1x1 matrix/vector. If A and B are 2x2 matrix, and v a 2x1 column vector, then A(Bv) = B(Av). None of the above.

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00:01 For this question, we are looking at three statements about matrix vector products, and we have to decide which, if any of them, is false.

00:10 So let's look at the first statement.

00:13 We're given a matrix a, which is m by n.

00:19 A little messy, sorry.

00:22 A is m by n, and v is n by 1.

00:30 So the product is defined.

00:32 Remember that if you have two matrices of size, let's say, a by b, and you're taking the product with a matrix of size b times c, then this gives you a matrix of size a times c.

00:53 And this product is only defined when these two numbers here agree.

01:00 That's why they're both b in this case.

01:02 So now we're looking at a times v.

01:07 So a times v, well, if we write this out in its full matrix form, we could write this as a11, a12, and so on up to a1n.

01:25 And then the next row will be a21, a22, up to a2n.

01:32 And then this continues down to the mth row am 1, am 2, which continues to a .mn.

01:47 And similarly, the vector v.

01:51 V2 .com, v .m.

01:52 V .2 dot, dot, v .n.

01:57 Now, if we carry out this matrix multiplication, what will it look like? excuse me, what will it look like? well, we know how to matrix multiply.

02:07 We take a1112 times v1, a12 times v2, and so on all the way up to the last entry for the first entry of our vector...

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