I am taking a signals processing class online and we are covering some basic linear algebra. Here is a question that I'm struggling with from the class.
Find a matrix $A$ which satisfies all of the following characteristics:
- $\text{dim }\text{col}(A) = 2$
- $\begin{bmatrix} 1 & 1 & 2 & -1\end{bmatrix}^T \in \text{null}(A)$
- $\begin{bmatrix} 3 & 3 & 6 \end{bmatrix}^T \in \text{col}(A)$
- $\begin{bmatrix} 3 & 0 & 0 & 3\end{bmatrix}^T \in \text{col}(A^T)$
Attempt
So far, I've started by making the first two columns of my matrix $A$ independent.
$$\begin{bmatrix} 1 & 0 & \alpha_1 & \beta_1 \\ 1 & 0 & \alpha_2 & \beta_2 \\ 0 & 1 & \alpha_3 & \beta_3 \end{bmatrix}$$
From this I can clearly fulfill the requirement that $\begin{bmatrix} 3 & 3 & 6 \end{bmatrix}^T \in \text{col}(A)$. Then I start filling in the $\alpha_i$ and $\beta_i$ values to fulfill the other characteristics.
The problem is, whenever I manage to satisfy a couple of the characteristics, I break the third one. I know how to individually satisfy these characteristics, but not how to satisfy them together. How can I go about constructing a matrix that satisfies all these characteristics?