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)=0_{v}and Im(L)=W; let (v_{1},v_{2 ,}v_{3) }be a basis of V; determine whether or not (L(v_{1}) L(v_{2}) L(v_{3})) 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∈M_{n}(R) be a real n x n matrix (where n is an integer greater than or equal to 2), and assumeλ_{1},λ∈ R are two eigenvalues of A with_{2 }λ_{1}does not equal(i.eλ_{2 }λare two distinct eigenvalues of A). let W_{1},λ_{2 }_{1}denote the eigenspace of A corresponding with eigenvalueλand let W_{1},_{2}denote the same with.λ_{2}

Show that we have W_{1}∩ W_{2}= {0_{Rn}}

I know that the W_{1}is simply the kernel of the matrix (Iλbut don't know how to get past that._{1}-A),