I know the answer to part a. Can you help me in solving parts b and c?
(a) Let $S=\{Ax|x\in R^n\}$, where $A$ is a given matrix. Show that $S$ is a subspace of $R^n$.
(b) Assume that $S$ is a proper subspace of $R^n$. Show that there exists a matrix $B$ such that $S = \{y \in R^n | By = 0\}$. Hint: Use vectors that are orthogonal to $S$ to form the matrix $B$.
(c) Suppose that $V$ is an $m$-dimensional affine subspace of $R^n$, with $m < n$. Show that there exist linearly independent vectors $a_1, ... , a_{n-m}$, and scalars $b_1, . . . , b_{n-m}$, such that $V = \{y | a'_iy = b_i, i = 1, ... , n - m\}$.
As you did not show much effort, I won't either.
Hint: If $S^\bot=\langle v_1,\dots,v_m\rangle$ then $$B=\begin{pmatrix}v_1^T\\\vdots\\v_m^T\end{pmatrix}$$ is a solution for (b). Up to you to figure out why.
For (c), use (b) and set $b_i=0$.