Linear mapping/Dimension formula/Fact/Proof

Set ${\displaystyle {}n=\dim _{K}{\left(V\right)}}$. Let ${\displaystyle {}U=\operatorname {kern} \varphi \subseteq V}$ denote the kernel of the mapping, and let ${\displaystyle {}k=\dim _{K}{\left(U\right)}}$ denote its dimension (${\displaystyle {}k\leq n}$). Let

${\displaystyle u_{1},\ldots ,u_{k}}$

be a basis of ${\displaystyle {}U}$. Due to fact, there exist vectors

${\displaystyle v_{1},\ldots ,v_{n-k}}$

such that

${\displaystyle u_{1},\ldots ,u_{k},\,v_{1},\ldots ,v_{n-k}}$

is a basis of ${\displaystyle {}V}$. We claim that

${\displaystyle w_{j}=\varphi (v_{j}),j=1,\ldots ,n-k,}$

is a basis of the image. Let ${\displaystyle {}w\in W}$ be an element of the image ${\displaystyle {}\varphi (V)}$. Then there exists a vector ${\displaystyle {}v\in V}$ such that ${\displaystyle {}\varphi (v)=w}$. We can write ${\displaystyle {}v}$ with the basis as

${\displaystyle {}v=\sum _{i=1}^{k}s_{i}u_{i}+\sum _{j=1}^{n-k}t_{j}v_{j}\,.}$

Then we have

${\displaystyle {}{\begin{aligned}w&=\varphi (v)\\&=\varphi {\left(\sum _{i=1}^{k}s_{i}u_{i}+\sum _{j=1}^{n-k}t_{j}v_{j}\right)}\\&=\sum _{i=1}^{k}s_{i}\varphi (u_{i})+\sum _{j=1}^{n-k}t_{j}\varphi (v_{j})\\&=\sum _{j=1}^{n-k}t_{j}w_{j},\end{aligned}}}$

which means that ${\displaystyle {}w}$ is a linear combination in terms of the ${\displaystyle {}w_{j}}$. In order to prove that the family ${\displaystyle {}w_{j}}$, ${\displaystyle {}j=1,\ldots ,n-k}$, is linearly independent, let a representation of zero be given,

${\displaystyle {}0=\sum _{j=1}^{n-k}t_{j}w_{j}\,.}$

Then

${\displaystyle {}\varphi {\left(\sum _{j=1}^{n-k}t_{j}v_{j}\right)}=\sum _{j=1}^{n-k}t_{j}\varphi {\left(v_{j}\right)}=0\,.}$

Therefore, ${\displaystyle {}\sum _{j=1}^{n-k}t_{j}v_{j}}$ belongs to the kernel of the mapping. Hence, we can write

${\displaystyle {}\sum _{j=1}^{n-k}t_{j}v_{j}=\sum _{i=1}^{k}s_{i}u_{i}\,.}$

Since this is altogether a basis of ${\displaystyle {}V}$, we can infer that all coefficients are ${\displaystyle {}0}$, in particular, ${\displaystyle {}t_{j}=0}$.