I want to prove the existence of complement subspace using the notion of quotient spaces, i.e. let $W \subseteq V$ be a subspace, then there is a subspace $U \subseteq V$ such that $V = W \oplus U$.
Say $W \subseteq V$ has basis $\{w_1, \cdots, w_m\}$, then $V / W$ is a vector space. Let $\{x_1, \cdots, x_n\}$ be a basis of $V /W$. Lift it back to $V$ using the inverse projection, i.e. let $u_i = v \in V$ such that $v + W = x_i$, for $i = 1, \cdots, n$. Then I think it makes intuitive sense that $\{w_1, \cdots, w_m, u_1, \cdots, u_n\}$ is a basis of $V$. But I don't quite see how I can show that rigorously. I guess it has something to do with how the projection map is set up. Any hint would be greatly appreciated. Thanks in advance!
You'll need to use that the images of the vectors $u_1, \dots, u_n$ under the projection map $\pi \colon V \to V/W$ form a basis, together with the fact that any element $v \in V$ is mapped to zero by $\pi$ if and only if $v \in W$, i.e. if and only if $v$ is a linear combination of $w_1, \dots, w_n$.
Let's see how to use this to prove $\{ w_1, \dots ,w_m, u_1, \dots ,u_n\}$ spans $V$. Let $v \in V$, then since $\{ x_1, \dots, x_n \}$ is a basis of $V/W$ we have $$ \pi(v) = \mu_1 x_1 + \cdots + \mu_n x_n $$ for some scalars $\mu_1, \dots, \mu_n$. Now notice that $\pi(\mu_1 u_1 + \cdots + \mu_n u_n)$ equals the sum above, so $v - (\mu_1 u_1 + \cdots + \mu_n u_n) \in \ker(\pi) = W = \text{span}\{ w_1, \dots, w_n \}$. So then $$ v - (\mu_1 u_1 + \cdots + \mu_n u_n) = \lambda_1 w_1 + \dots + \lambda_n w_n $$ for some scalars $\lambda_1, \dots, \lambda_n$. This shows $v$ is a linear combination of $\{ w_1, \dots ,w_m, u_1, \dots ,u_n \}$ , and so this is a spanning set.
The proof that it is a linear independent set is similar: given some linear dependence relation, consider the image in $V/W$ and use that the elements $\pi(u_i) = x_i$ (where $i = 1, \dots, n$) form a basis of $V/W$. Can you complete this proof yourself?