I have a practice problem I attempted to attempt, but got very stuck. I mostly get how to make operators when it's not in a matrix, however the moment I'm switching to matrices I get lost.

how would I go about creating an R operator in the space of 2x2 matrices without any 1-dimensional R-invariant subspaces? It should have two 2-dimensional subspaces X and Y? X and Y's direct sum is the whole space.

I'm not following any thing above, but my notes said it's useful to take note of the fact that U and W are subspaces of V, and U + W would be R-invariant even if U and W aren't.

even if someone explained how to do this in simple terms I would be greatful. Thanks.