We have these two matrices:
$K = \left(\begin{matrix} 2 & 1 \\ 8 & 7\end{matrix}\right), \quad L = \left(\begin{matrix} 2 & 1 \\ 2 & 7 \end{matrix} \right)$
We have been asked if every matrix of $\mathbb{R}^{2 \times 2}$ can be written as a linear combination of $K$ and $L$ matrices. This means that the set $\{K,L\}$ is a base of $\mathbb{R}^{2 \times 2}$, right?
I've thought of this: For $K$ and $L$ matrices to be a base of $\mathbb{R}^{2 \times 2}$ they must be linearly independent, is that correct?
$a,b$ numbers of $\mathbb{R}$
$a \cdot K + b \cdot L = 0$, where $0$ is the $\left(\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}\right)$ matrix.
So:
$\begin{array}{cccc} 2a &+& 2b &=& 0 \\ a &+& b &=& 0 \\ 8a &+& 2b &=& 0 \\ 7a &+& 7b &=& 0 \end{array}$
(the solution set of this system is empty set?)
How can I think of that?
Thank you!