Linear mapping/Change of field/Matrix/Fact

Let $K$ be a field, and let $V$ be an $n$ -dimensional $K$ -vector space, endowed with a basis $𝔳 = v_{1}, \dots, v_{n}$ , and let $W$ be an $m$ -dimensional $K$ -vector space, endowed with a basis $𝔴 = w_{1}, \dots, w_{m}$ . Let

φ : V ⟶ W

be a linear mapping with the describing matrix $M = M_{𝔴}^{𝔳} (φ) = (a_{i j})$ . Let $K \subseteq L$ be a field extension.

Then the linear mapping

φ_{L} : V_{L} ⟶ W_{L}

given by extension of scalars

is also described by the matrix

M

, considered over

L

, with respect to the bases

1 \otimes v_{1}, \dots, 1 \otimes v_{n}

of

V_{L}

and

1 \otimes w_{1}, \dots, 1 \otimes w_{m}

of

W_{L}

.

Proof, Write another proof