Linear mapping/Finite dimensional/Eigenvalues bounded by dimension/Fact

Let ${}K$ be a field, ${}V$ a finite-dimensional ${}K$-vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.

Then there exist at most ${}\dim _{K}{\left(V\right)}$ many

eigenvalues

for ${}\varphi$.

Write a proof