Change of Basis Matrix under Linear Transformation/Corollary

Corollary to Change of Basis Matrix under Linear Transformation

Let $R$ be a commutative ring with unity.

Let $G$ be a free unitary $R$-module of finite dimension $n$.

Let $\sequence {a_n}$ and $\sequence { {a_n}'}$ be ordered bases of $G$.

Let $u: G \to G$ be a linear operator on $G$.

Let $\sqbrk {u; \sequence {a_n} }$ denote the matrix of $u$ relative to $\sequence {a_n}$.

Let:

$\mathbf A = \sqbrk {u; \sequence {a_n} }$

$\mathbf B = \sqbrk {u; \sequence { {a_n}'} }$

Then:

$\mathbf B = \mathbf P^{-1} \mathbf A \mathbf P$

where $\mathbf P$ is the matrix corresponding to the change of basis from $\sequence {a_n}$ to $\sequence { {a_n}'}$.

Proof

This is an instance of Change of Basis Matrix under Linear Transformation.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices: Theorem $29.4$: Corollary