Linear mapping/Finite dimensional/Change of basis/Fact

Let ${}K$ denote a field, and let ${}V$ and ${}W$ denote finite-dimensional ${}K$ -vector spaces. Let ${}{\mathfrak {v}}$ and ${}{\mathfrak {u}}$ be bases of ${}V$ , and ${}{\mathfrak {w}}$ and ${}{\mathfrak {z}}$ bases of ${}W$ . Let

\varphi \colon V\longrightarrow W

denote a linear mapping, which is described by the matrix ${}M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )$ with respect to the bases ${}{\mathfrak {v}}$ and ${}{\mathfrak {w}}$ .

Then

{}\varphi

is described with respect to the bases

${}{\mathfrak {u}}$ and ${}{\mathfrak {z}}$ by the matrix

M_{\mathfrak {z}}^{\mathfrak {w}}\circ (M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))\circ (M_{\mathfrak {u}}^{\mathfrak {v}})^{-1},

where ${}M_{\mathfrak {u}}^{\mathfrak {v}}$ and ${}M_{\mathfrak {z}}^{\mathfrak {w}}$ are the transformation matrices that describe the change of basis from ${}{\mathfrak {v}}$ to ${}{\mathfrak {u}}$ and from

{}{\mathfrak {w}}

to

{}{\mathfrak {z}}

.

Proof, Write another proof