Linear algebra problem involving image spaces

1. Oct 1, 2012

Sandbox.WeC

1. The problem statement, all variables and given/known data
A is a mxn. V is nxn and invertible. Show that imA=imAV

2. The attempt at a solution
Up until now I haven't done much in the way of proving things. In this case is it enough to show that they are each closed under addition and scalar multiplication? Would that mean that imA is in imAV and vice versa, meaning they are equal?

Thanks for any help

Last edited: Oct 1, 2012
2. Oct 1, 2012

HallsofIvy

Since A is "m by n", A maps Rn to Rm. Since V is "n by n", V maps Rn to Rn[/b] so that AV maps Rn to Rm.

Suppose w is in Im(A). Then there exist v in Rn such that Av= w. Since V is invertible, there exist u in Rn such that Vu= v. Then (AV)u= A(Vu)= A(v)= w. That is Im(AV) is a subset of Im(V). To prove that Im(AV)= Im(A), you must prove that Im(A) is a subset of Im(AV). To do that start "suppose w is in Im(AV). I will leave it to you.

3. Oct 1, 2012

Sandbox.WeC

I see where you are going with that. Does that mean that everything I did was wrong? I thought that if I proved they were subspaces of each other that would mean they are equal