Linear algebra problem involving image spaces

[Sandbox.WeC]

Homework Statement

A is a mxn. V is nxn and invertible. Show that imA=imAV2. 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

 

[HallsofIvy]

Since A is "m by n", A maps Rⁿ to R^m. Since V is "n by n", V maps Rⁿ to R<sup>n[/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.</sup>

 

[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