<s>[INST] How do I determine if a matrix is in a span of two other matrices? 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}
 [/INST] 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 $κ\cdot1+λ\cdot1=1$. 
</s>