00:01
For this question, you've been asked, let v be the set of all pairs x, y of real numbers with addition defined component -wise and scalar multiplication also defined component -wise.
00:18
You've been asked to verify all the properties of vector space with respect to these operations.
00:25
So here's a list of the properties that we need to verify.
00:41
So the first one is associativity of vector addition.
00:49
So we'll take three vectors x1, y1 plus x2, y2 plus 3, y3.
01:23
That's equal to x1 y1 plus x2 plus x3.
01:44
And we've got y2 plus y3 in the third component.
01:54
And that's equal to x1 plus x2 plus x3.
02:09
Y1 plus y2 plus y3 and if we were to add the first two vectors together first we have x1 y1 plus x2 y2 and then plus x3 y3 and that's equal to x1 plus x2, y1 plus y2 plus x3, y3.
03:02
That's equal to x1 plus x2 plus x3 y1 plus y2 plus y3 and that's the same as what we got before so that shows that the addition is associative next we need to show commutativity so so x1, y1 plus x2, y2.
03:50
We're adding component y, so we get x1 plus x2 and y1 plus y2...