Let $V$ and $W$ be vector spaces over $F$, where $V$ is $n$-dimensional. Let $K\subseteq V$ and $R\subseteq W$ be finite-dimensional subspaces such that $\dim K+\dim R=n$. Prove that there exists a linear transformation $L : V \to W$ such that $\ker L=K$ and $\text{Im}\,L=R$.
My work;
Let $\dim(V)=m$ and the basis of $V=\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$
Let $\dim(W)=n$ and the basis of $W=B_{1}, B_{2}, \cdots B_{n}$
Let $\dim(K)=d$ and the basis of $K=a_{1}, a_{2}, \ldots a_{d}$
Let $\therefore \operatorname{dim}(R)=m-d$ and we can take its basis as
$$
\beta_{1}, \beta_{2} \cdots \cdot \beta_{m-d} .
$$
Best Answer
The fact: Any basis for $K$ can be extended to a basis for $V$.
Let $\dim(K)=d$ and the basis of $K$ be $\{\alpha_1,\cdots,\alpha_d\}$, then $\{\alpha_1,\cdots ,\alpha_d,\alpha_{d+1},\cdots,\alpha_n\}$ is a basis of $V$.
Let $\dim(R)=n-d$ and the basis of $R$ be $\{\beta_{d+1},\cdots,\beta_{n}\}$, then $\{\beta_1,\cdots ,\beta_d,\beta_{d+1},\cdots,\beta_n\}$ is a basis of $W$.
Define the map $L$： $$L(\alpha_i)=0,\text{for}\ \ \ 1\le i\le d,$$ $$L(\alpha_i)=\beta_i,\text{for}\ \ \ d+1\le i\le n.$$
Let $L$ be linear, then we prove it.