"Show that V, the set of all square matrices whose trace is 0, form a subspace of $M_{n,n}$ (the set of all square matrices). What is its dimension?"
I have shown that $V$ is a subspace, but I do not know how to determine its dimension yet alone what a base of $V$ would possibly look like. Hints and solutions are appreciated!
Thanks.
Best Answer
Trace is a nonzero linear function $M_{n,n}\to\Bbb R$, so it has rank$~1$ (since at arrival the space has dimension$1$, it cannot be more than that). Now use the rank-nullity theorem.