Given $V$ and $W$ a finite-dimensional spaces, let $U$ be a vector space of $V$

Question

i need to prove there is a linear mapping $T:V\to W$ so that $ker~T= U$ if and only if $\dim U\geq \dim V - \dim W$.

Answer 1

The rank-nullity theorem tells you that for such a linear map $$ \dim\ker T+\dim\operatorname{im}T=\dim V $$ that is, $$ \dim\operatorname{im}T=\dim V-\dim U $$ In particular $\dim V-\dim U\le\dim W$.

Conversely, assume that $$ \dim V-\dim U\le\dim W $$ Take a basis $\{v_1,\dots,v_k\}$ of $U$ and extend it to a basis $\{v_1,\dots,v_k,v_{k+1},\dots,v_n\}$ of $V$. Take a basis $\{w_1,\dots,w_m\}$ of $W$ and note that $n-k\le m$. Can you define a linear map with the required properties on the basis of $V$?