(Solved): explain this example how k=3 and 4 EXAMPLE 5 Let A=223457278136 ...

EXAMPLE 5 Let $A=\left[\begin{array}{rrrr}2& 4& -2& 1\\ -2& -5& 7& 3\\ 3& 7& -8& 6\end{array}\right]$ a. If the column space of $A$ is a subspace of ${\mathbb{R}}^k$, what is $k$ ? b. If the null space of $A$ is a subspace of ${\mathbb{R}}^k$, what is $k$ ? SOLUTION a. The columns of $A$ each have three entries, so $\mathrm{Col}A$ is a subspace of ${\mathbb{R}}^k$, where $k=3$ b. A vector $\mathrm{x}$ such that $A\mathrm{x}$ is defined must have four entries, so Nul $A$ is a subspace of ${\mathbb{R}}^k$, where $k=4$.