Relative Matrix of Composition of Linear Transformations
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Theorem
Let $R$ be a ring with unity.
Let $M, N, P$ be free $R$-modules of finite dimension $m, n, p > 0$ respectively.
Let $\AA, \BB, \CC$ be ordered bases of $M, N, P$.
Let $f: M \to N$ and $g : N \to P$ be linear transformations, and $g \circ f$ be their composition.
Let $\mathbf M_{f, \BB, \AA}$ and $\mathbf M_{g, \CC, \BB}$ be their matrices relative to $\AA, \BB$ and $\BB, \CC$ respectively.
Then the matrix of $g \circ f$ relative to $\AA$ and $\CC$ is:
- $\mathbf M_{g \mathop \circ f, \CC, \AA} = \mathbf M_{g, \CC, \BB} \cdot \mathbf M_{f, \BB, \AA}$
Proof 1
Let $m \in M$, and $\sqbrk m_\AA$ be its coordinate vector with respect to $\AA$.
On the one hand:
\(\ds \sqbrk {\map g {\map f m} }_\CC\) | \(=\) | \(\ds \mathbf M_{g \mathop \circ f, \CC, \AA} \cdot \sqbrk m_\AA\) | Change of Coordinate Vectors Under Linear Mapping applied to $g \circ f$ |
On the other hand:
\(\ds \sqbrk {\map g {\map f m} }_\CC\) | \(=\) | \(\ds \mathbf M_{g, \CC, \BB} \cdot \sqbrk {\map f m}_\BB\) | Change of Coordinate Vectors Under Linear Mapping applied to $g$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf M_{g, \CC, \BB} \cdot \mathbf M_{f, \BB, \AA} \cdot \sqbrk m_\AA\) | Change of Coordinate Vectors Under Linear Mapping applied to $f$ |
Thus:
- $\forall m \in M: \paren {\mathbf M_{g \mathop \circ f, \CC, \AA} - \mathbf M_{g, \CC, \BB} \cdot \mathbf M_{f, \BB, \AA} } \cdot \sqbrk m_\AA = 0$
The result follows.
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$\blacksquare$
Proof 2
Let:
- $\AA = \sequence {a_m}$
- $\BB = \sequence {b_n}$
- $\CC = \sequence {c_p}$
Let:
- $\sqbrk \alpha_{m n} = \sqbrk {f; \sequence {b_n}, \sequence {a_m} }$
and:
- $\sqbrk \beta_{n p} = \sqbrk {g; \sequence {c_p}, \sequence {b_n} }$
Then:
\(\ds \map {\paren {g \circ f} } {a_j}\) | \(=\) | \(\ds \map v {\map f {a_j} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map v {\sum_{k \mathop = 1}^n \alpha_{k j} b_k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \alpha_{k j} \map v {b_k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \alpha_{k j} \paren {\sum_{i \mathop = 1}^p \beta_{i k} c_i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \paren {\sum_{i \mathop = 1}^p \alpha_{k j} \beta_{i k} c_i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^p \paren {\sum_{k \mathop = 1}^n \alpha_{k j} \beta_{i k} c_i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^p \paren {\sum_{k \mathop = 1}^n \alpha_{k j} \beta_{i k} } c_i\) |
$\blacksquare$
Also see
- Product of Change of Basis Matrices, an analogous result for change of basis
- Composition of Linear Transformations is Linear Transformation