Give an example of a linear operator $\displaystyle L $on $\displaystyle M_{2x3}(F)$ such that $\displaystyle N(L) = R(L)$
Is it true that $\displaystyle M_{2\times 3}(F)$ is just the set of all $\displaystyle 2\times 3$ matrices over the field $\displaystyle F?$ And $\displaystyle N(L)$ is the kernel (the set of all vectors in the domain that get mapped to the zero vector in the range), and $\displaystyle R(L)$ is the range of $\displaystyle L?$ Because, if all of this is true, your problem is a trick question. It's impossible to answer, because the kernel is a subset of $\displaystyle F^{3},$ whereas the range is a subset of $\displaystyle F^{2}.$ I suppose you could think of the equals sign as a sort of embedding, but that's not very usual.