How do I determine if a matrix is in a span of two other matrices?

Question

If \begin{align} A =\begin{bmatrix} 1 & 1 \\ -1 & 1 \\ \end{bmatrix} \end{align} and \begin{align}B = \begin{bmatrix} 1 & -1 \\ 1 & 0 \\ \end{bmatrix} \end{align}

So, how can I check if $C$ is in span $(A,B)$ when \begin{align}C= \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \end{align}

Answer 1

Hint: If you can find $κ, λ$ such that $$κ\begin{pmatrix}1 & 1\\-1&1 \end{pmatrix}+λ\begin{pmatrix}1 & -1\\1&0 \end{pmatrix}=\begin{pmatrix}1 & 2\\3&4 \end{pmatrix}$$ then, the answer is yes. You have $4$ equations with $2$ unknowns, the first of the equations being $κ\cdot1+λ\cdot1=1$.

Answer 2

The space of $2\times 2$ matrices is simply $\Bbb R^4$. Form a matrix with the vectors $(1,1,-1,1)$, $(1,-1,1,0)$ and $(1,2,3,4)$ and check its rank.