00:01
So here we have two linear transformations.
00:04
We have three vector spaces, v, w, and u.
00:12
T is a transformation from v to w, and l from w to u.
00:16
We want to know is l compose t a linear transformation from v to u? i'm going to claim that yes, it is.
00:27
Let's see.
00:28
So we'll let v1 and v2 be in v.
00:35
Then l compose t of v1 plus v2 is l of t of v1 plus v2.
00:47
Let's even give it some scalars here just to do everything at once.
00:54
It's alpha v1 plus beta v2.
00:56
That's l of t of alpha v1 plus beta v2.
01:01
By linearity, that's going to be l of alpha t of v1 plus alpha t of v1 plus beta times t of v2, which again by linearity of l is going to be alpha times l of t of v1 plus beta times l of t of v2.
01:33
That is alpha l compose t of v1 plus beta l compose t of v2.
01:43
Great.
01:45
So the composite is in fact linear.
01:49
Yes, it is.
01:52
If t and l are isomorphisms, is l compose t also an isomorphism? well, we know that it's linear, so all we need to do is consider if it's bijective...
