Good morning. I am a little curious about the motivation for a simple example from linear algebra as well as critique on the example I came up with.
Question: Given a subspace of $M$ commuting $4 \times4$ matrices with complex entries give an example that shows $M$ has five linearly independent elements.
First here is the example I came up with is there a better one? $ \left( \begin{array} 01 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) \left( \begin{array} 00 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) \left( \begin{array} 00 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) \left( \begin{array} 00 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} 00 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array} \right) $
I think the dimension of $M$ is bounded by 4 when there exists a matrix in $M$ with two distinct characteristic values but is that the only reason to construct such an example from a teaching perspective?