Column space, row space, and null space of a matrix

In this exercise, we use scalars from the field $\mathbb{Z}_{5}$ of integers modulo $5$ instead of real numbers (see Section an exercise given elsewhere in the book, “Fields”). Determine the rank and nullity and find a basis of the column space, row space, and null space of the following matrix over $\mathbb{Z}_{5}$.

Show that if $A$ is an $m\times n$-matrix, then $\operatorname{null}(A)$ is a subspace of $\mathbb{R}^{n}$.

Let $A$ be an $m\times n$-matrix. Show that $\operatorname{col}(A)=\left\{A\mathbf{u}\mid\mathbf{u}\in\mathbb{R}^{n}\right\}$.

Show that $\operatorname{rank}(A)=\operatorname{rank}(A^{T})$.

For invertible matrices $B$ and $C$ of appropriate size, show that $\operatorname{rank}(A)=\operatorname{rank}(BA)=\operatorname{rank}(AC)$.

Suppose $A$ is an $m\times n$-matrix and $B$ is an $n\times p$-matrix. Show that

Hint: Consider the subspace $\operatorname{col}(B)\cap\operatorname{null}(A)$ and suppose a basis for this subspace is $\left\{\mathbf{w}_{1},\ldots,\mathbf{w}_{k}\right\}$. Let $\left\{\mathbf{z}_{1},\ldots,\mathbf{z}_{k}\right\}$ be such that $B\mathbf{z}_{i}=\mathbf{w}_{i}$. Now suppose $\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{r}\right\}$ is a basis for $\operatorname{null}(B)$, and argue that $\operatorname{null}(AB)\subseteq\operatorname{span}\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{r},\mathbf{z}_{1},\ldots,\mathbf{z}_{k}\right\}$.