Let A be an m x n matrix with entries in R. Let T_A : R^n -> R^m be the linear map T_A(X) = A_X. Let U be the solution set of the homogeneous linear system A_X = O. Let W be the set of all vectors Y such that Y = A_X for some X in R^n. I don't really know what I'm supposed to do here, any help would be greatly appreciated.

Let A be an m x n matrix with entries in R. Let T_A : R^n -> R^m be the linear map T_A(X) = A_X. Let U be the solution set of the homogeneous linear system A_X = O. Let W be the set of all vectors Y such that Y = A_X for some X in R^n. I don't really know what I'm supposed to do here, any help would be greatly appreciated.

Let's get our definitions straight:
\begin{aligned}
U &= \text{Ker } A \\
W &= \text{Im } A
\end{aligned}

According to the isomorphism theorem (see e.g. wiki), $\text{Im }A$ is isomorphic with the quotient space $\mathbb R^n / \text{Ker }A$.$\qquad \blacksquare$