$X_0\subset X$ finite codimension. Show there exists subspace $Z$ s.t. $X= Z\oplus X_0$

Question

I'm not so familiar with these types of arguments concerning vector spaces, and direct sums.

Let $X$ be a vector space and $X_0\subset X$ with finite codimension, i.e. $dim(X/X_0)$ is finite.

Show there exists a subspace $Z\subset X$ with $dim(Z)=dim(X/X_0)$ and $X=Z\oplus X_0$.

(so for all $x\in X$ there exists unique $x'\in X_0$ and $x''\in Z$ such that $x=x'+x''$)

Some light on this is very much appreciated.

Answer 1

Let $[z_1]$, $[z_2]$, $\ldots$, $[z_n]$ be a basis of $X/X_0$. Then consider $Z:=\operatorname{span}\{z_1, z_2, \ldots, z_n\}$.