Linear mapping/Dimension formula/Fact

Dimension formula

Let ${}K$ denote a field, let ${}V$ and ${}W$ denote ${}K$-vector spaces, and let

$\varphi \colon V\longrightarrow W$

denote a ${}K$-linear mapping. Suppose that ${}V$ has finite dimension.

Then

${}\dim _{}{\left(V\right)}=\dim _{}{\left(\operatorname {kern} \varphi \right)}+\dim _{}{\left(\operatorname {Im} \varphi \right)}\,$

holds.

Write a proof