Let V be an n-dimensional vector space, and let $0 ≤ m < n$. If W is an $m$-dimensional subspace of V and X is an ($n-m$)-dimensional subspace of V, show that there exists a linear transformation $T: V{\longrightarrow}V$ such that ker (T) = W and Im (T) = X.
I don't really know how to start the proving. I can't grasp the idea of (n-m)-dimensional subspace. I wish you can give me clues on how can I approach this problem. Thank you in advance!
Best Answer
Choose a basis $w_1,\dots,w_m$ in $W$ and extend it to a basis $w_1,\dots,w_m,\ u_1,\dots,u_{n-m}$ of $V$.
Also choose a basis $x_1,\dots,x_{n-m}$ of $X$.
Finally, consider the linear extension of the mapping $$w_i\mapsto 0\,,\\ u_j\mapsto x_j\,.$$