00:01
In this problem, we have a set v defined as given.
00:05
So v is the set of 3 by 3 matrices with only diagonal elements that are not zero, and those diagonal elements are real numbers.
00:16
So this is how i just denoted that here.
00:20
What we want to show in this problem is that this set v is a vector space.
00:29
So in order to do that, what we need to do is show that this thing obeys the e -axiums of a vector space.
00:39
Now, all of these axioms are going to be generally, obviously true, but since we need to show this, we need to be thorough, essentially.
00:50
So let's go through these.
00:52
So the first one is associativity of addition.
01:01
I'll just abbreviated as add.
01:05
So what does this say? well, this says that if we have a summation of three elements, x plus y plus c, that there's two ways we can add this, essentially.
01:17
We can add this first, or we can add the second two first.
01:22
And this is obviously true.
01:25
This is a property of matrix multiplication.
01:34
Or no multiplication, well, matrix addition.
01:40
It is also for multiplication, but that's not the point here.
01:45
And so, just by the fact that matrices added in this way anyways, you know, it doesn't matter if they are diagonal or not, this holds.
01:56
So on to the second one, let's see, it's a commutivity of addition as well.
02:07
Now this is another one that's obvious.
02:11
It's the order in which you add things.
02:15
Whoops, i should be next.
02:18
The order that which you add things doesn't matter.
02:20
And so we can say the same thing here, right? it's also a property of matrix edition that it doesn't matter which order that you add.
02:33
Alright, next one, identity element.
02:43
Element of addition as well.
02:46
So the identity element of addition is simply, in this case, 0 ,000, 0 ,000, 0 ,000, right? and this just corresponds to when you pick a is equal to b is equal to c is equal to 0, right? and so this is of course an element of v, and thus our set contains the identity element of addition.
03:14
Okay, fourth axiom, inverse elements of addition.
03:23
So this is to say that if you have a matrix a -b -c 0 -0 -0 -0, so on and so forth, there is another matrix in the set v, which if you add it to this, you get the identity element, this one.
03:43
So we pick this matrix in v, right? and what this means also is that you also have negative a, negative b, negative c.
03:56
And v, right? and this just comes from the fact that if a, b, c, are real numbers, then the negative of them are real numbers...