This is not homework, class or a project. I've been out of college for some time now and decided to learn math on my own time.
I can't figure out how to solve the following problem:
Construct a 4 x 4 matrix A whose column space R and null space N are given by
$$ R = \alpha \begin{bmatrix} 1\\2\\0\\0 \end{bmatrix} + \beta \begin{bmatrix} 0\\1\\2\\0 \end{bmatrix}$$
$$ N = \alpha \begin{bmatrix} 1\\2\\0\\0 \end{bmatrix} + \beta \begin{bmatrix} 0\\1\\2\\0 \end{bmatrix}$$
How do I approach this problem?
Best Answer
Let $A=[v_1\;v_2\;v_3\;v_4]$ be the required matrix. You need that $$ A\begin{bmatrix}1\\2\\0\\0\end{bmatrix}=v_1+2v_2=0 $$ so $v_1=-2v_2$; also $$ A\begin{bmatrix}0\\1\\2\\0\end{bmatrix}=v_2+2v_3=0 $$ so $v_2=-2v_3$ and $v_1=4v_3$.
You see that $v_4$ can be anything; now just take $$ v_3=\begin{bmatrix}1\\2\\0\\0\end{bmatrix} \quad v_4=\begin{bmatrix}0\\1\\2\\0\end{bmatrix} $$ Then the matrix $$ A=[v_1\;v_2\;v_3\;v_4]= \begin{bmatrix} 4 & -2 & 1 & 0 \\ 8 & -4 & 2 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ will have the required column space and null space: indeed, the column space is generated by $v_3$ and $v_4$, so it has dimension $2$. Since the null space contains the required one, it will be equal to it by the rank-nullity theorem.