OK did the second one, can someone help me with 1B) please!
Hi, I have an assignment due and I have done most of the questions there is just one part of a question I have left, if someone can help that would be amazing
Edit: I answered the second question, now all I need help with is 1B).
I already did the first question part A), but I can't figure out B
Thanks
For the first question part B, I know all the vectors in B are independent because B is a basis for W, so I know the inverse of all these vectors are also independent. And I know that A maps the vectors in U onto W, so the inverse of A will map the vectors in W, or B, onto U...but how should I write this out?
Something obviously happened to your last post!
The one you are having difficulty with is, I assume, 3b since there is no "problem 1" given. It says:
If A is an n by n invertible matrix from U to W and is a basis for W, then is a basis for U. Since A is n by n, both U and W must be of dimension n so we need only show that is independent. Suppose for some scalars, , , , . Apply to A to both sides of that equation. What does that, together with the fact that [tex]\{v_1, v_2, \cdot\cdot\cdot, v_n\}[tex] is a basis, tell you about the scalars?
EDIT: First of all, as HallsofIvy pointed out, your notation is off, here. You need to prove that is a basis for , not . That said....
There are a couple of ways to do this. Going by HallsofIvy's method, we need to show that is linearly independent. To do this, let
for some scalars .
Multiplying both sides by , we have
.
But since is linearly independent, then for each , which is to say is linearly independent, and the conclusion follows.