Construct matrix A with two given vectors, c,d where Col A = Nul A = Span {c,d}

Question

Let c = $\begin{bmatrix}1 \\2 \\1 \\ 2\end{bmatrix}$ and let d =$\begin{bmatrix}1 \\0 \\0 \\ 1\end{bmatrix}$. Give an example of a matrix A with the property,

Span {c,d} = Nul A = Col A

So, I tried denoting A = [v1 v2 v3 v4] and solve the following equations:

A $\begin{bmatrix}1 \\2 \\1 \\ 2\end{bmatrix}$ = v1 + 2 v2 + v3 + 2 v4 = $0$ and

A $\begin{bmatrix}1 \\0 \\0 \\ 1\end{bmatrix}$ = v1 + v4 = $0$. Therefore v1 = -v4 , so c and d cannot be the first and fourth column,

since c and d are in Col A and thus in A.

So I found A= \begin{bmatrix} a11 & 1 & 1 & a14 \\ a21 & 2 & 0 & a24 \\ a31 & 1 & 0 & a34 \\ a41 & 2 & 1 & a44 \end{bmatrix}

Or A = \begin{bmatrix} a11 & 1 & 1 & a14 \\ a21 & 0 & 2 & a24 \\ a31 & 0 & 1 & a34 \\ a41 & 1 & 2 & a44 \end{bmatrix}

However, I am stuck and I do not know how to proceed, any suggestions?

Answer 1

You haven’t exhausted the constraints imposed by $\operatorname{Span}\{\mathbf c,\mathbf d\}=\operatorname{Nul}A$ yet. For either of the possibilities for $A$ that you’ve worked out so far, the equations $A\mathbf c=\mathbf 0$ and $A\mathbf d=\mathbf 0$ give you a system of 8 homogeneous linear equations in 8 unknowns. Solve them and you’ll have $A$.