00:01
So the first question is, is matrix multiplication commutative? does a -b always equal b -a? and the answer is false.
00:10
This is one of the major points of matrix multiplication, that it is not, in fact, commutative.
00:16
But you can show yourself as an example, something, even just a diagonal matrix, like, you know, 3 -01 times one -one here is going to equal.
00:29
Here.
00:30
Here, here, here we hit a three, here we hit again water falling down here, another three, and then here we hit a one and one, so we get a three three one from that, but one one one times three zero zero one.
00:55
Now here when we waterfall one and one times three and zero, again we get a three, but now one and one times the 0 and 1, we get a 1, and again, we're going to get a 3 into 1.
01:06
And these two matrices are not equal, and so this is indeed false.
01:12
All right.
01:13
So next, we want to consider two sets of functions.
01:17
So v is all functions that are solutions to f -double prime plus f -prime plus f -equal 0, and w are all functions that are solutions to f double prime plus f prime plus f equals x squared.
01:52
And we're asked which is a vector space.
01:55
And so functions, the addition and scalar multiplication and whatnot, that's all defined already for functions.
02:03
So we just, for this to be a vector space, we need to check that, 0 is an element, if f1 and f2 are elements, then f1 plus f2 is an element, and that any scalar lambda times f1 is an element.
02:24
And immediately we see that 0 equals x squared is not true.
02:32
And so we conclude that w is not a vector space.
02:38
And so now we just need to ask is v a vector space.
02:49
Okay, well, the derivative and second derivative of zero is zero, so zero plus zero plus zero equals zero is true...
