1. Lin alg proofs help

Hi everyone!

I'm having trouble with these two linear algebra proofs, and help would be greatly appreciated.

1. let V and W be real vector spaces, and let L: V to W be a linear transformation such that Ker(L)=0v and Im(L)=W; let (v1 ,v2 ,v3) be a basis of V; determine whether or not (L(v1) L(v2) L(v3)) is a basis for W

I know how to prove they are lin independent, but how do i prove that they're both lin independent and a generating set?

2. Let A∈Mn(R) be a real n x n matrix (where n is an integer greater than or equal to 2), and assume λ1, λ2 ∈ R are two eigenvalues of A with λ1does not equal λ2 (i.e λ1, λ2 are two distinct eigenvalues of A). let W1 denote the eigenspace of A corresponding with eigenvalue λ1, and let W2 denote the same with λ2 .
Show that we have W1 ∩ W2 = {0Rn}

I know that the W1 is simply the kernel of the matrix (I
λ1-A), but don't know how to get past that.

2. Re: Lin alg proofs help

Hey Tra003.

For a basis, you need N linearly independent vectors where dim(V) = N. Once you have this, you definitely have a basis for any vector space.

For the second one you should show that the two spaces are independent. If you are talking about two vectors then showing they are independent (i.e. not scalar multiples) of each other should suffice and you can do this by setting up a two row matrix and row-reducing it.

3. Re: Lin alg proofs help

Hi,
I think even with "easy" problems, it helps to see complete proofs: