Question

Does the following matrices span the space \[M _{2\times2}(\mathbb{R})\]:\[\left[\begin{matrix}1 & 2 \\ 0 & 0\end{matrix}\right],\left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right],\left[\begin{matrix}1 & 0 \\ 1 & 0\end{matrix}\right],\left[\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right]\]

Answer 1

I would say no because there is no way to get nonzero values for the element in the bottom right hand corner

Answer 2

how do i go about showing it though?

Answer 3

Use a counterexample. For instance, you cannot represent the matrix

\[\left[\begin{matrix}0 & 0 \\ 0 & 1\end{matrix}\right]\]

as a linear combo of the other four matrices given above

Answer 4

why? because all four matrices given have zeros in that bottom right hand corner. Any linear combo of all zeros will give you zero.

Answer 5

oh i see. all right. i get it now. thanks!