$T:V \to W, U: W \to Z$. $V, W, Z$ are vector spaces over $F$, then $U \circ T = UT:V \to Z$ is also a linear transformation.
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<span class="proof__expand"><a>[expand]</a></span>
$V$ is a vector space over a field $F$. Let $T, U_1, U_2 \in \L(V)$, then the following holds
- $T(U_1 + U_2) = TU_1 + TU_2$
- $(U_1 + U_2)T = U_1T + U_2T$
- $T(U_1U_2) = (TU_1)U_2$
- $TI = IT = T$(where $I$ is the identity linear transformation)
- $a(U_1U_2) = (aU_1)U_2 = U_1(aU_2), \forall a \in F$
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<span class="proof__expand"><a>[expand]</a></span>
Let $A $ be an $m \times n $ matrix and $B $ be an $n \times p $ matrix. We define the product of $A $ and $B $, denoted $AB $, to be the $m \times p $ matrix s.t.
$$(AB)_{ij} = \sum_{ k=1 }^{ n } A_{ik}B_{kj}, \text{ for } i \leq i \leq m, 1 \leq j \leq p$$
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<span class="proof__expand"><a>[expand]</a></span>
$T:V \to W, U: W \to Z$,
$\alpha, \beta, \gamma$ are ordered basis of $V, W$ and $Z$ respectively, then
$$[UT]^\gamma_\alpha = [U]^\gamma_\beta[T]^\beta_\alpha$$
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<span class="proof__expand"><a>[expand]</a></span>
Let $V $ be a finite-dimensional vector space with an ordered basis $\beta $. Let $T, U \in \L(V)$, then
$$[UT]_ \beta = [U]_ \beta [T]_ \beta $$
Let $V, W $ be two finite-dimensional vector spaces. $T: V \to W$. Let $\beta$ and $\gamma$ be ordered basis $V $ and $W $, respectively, then
$$[ T ]_{ \beta }^{ \gamma } [ v ]_{ \beta } = [ T(v) ]_{ \gamma }, \forall v \in V$$
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<span class="proof__expand"><a>[expand]</a></span>
Let $A $ be an $m \times n $ matrix with entries from a field $F $. We denote by $L_A $ the mapping $L_A: F^n \to F^m $ defined by $L_A(x) = Ax $(the matrix product of $A $ and $x $) for each column vector $x \in F^n $. We call $L_A $ a left-multiplication transformation.
Let $A $ be an $m \times n $ matrix with entries from $F $. Then the left-multiplication transformation $L_A: F^n \to F^m $ is linear. Furthermore, if $B $ is any other $m \times n $ matrix (with entries from $F $), and $\beta$ and $\gamma $ are the standard ordered bases for $F^n $ and $F^m $, respectively, then we have the following properties.
- $[L_A]^ \gamma _ \beta = A $
- $L_A = L_B $ if and only if $A = B $
- $L_{A+B} = L_A + L_B $ and $L_{aA} = a L_A $ for all $a \in F$.
- If $T : F^n \to F^m $ is linear, then there exists a unique $m \times n $ matrix $C $ s.t. $T = L_C $. In fact, $C = [ T ]_{ \beta}^{ \gamma } $
- If $E $ is an $n \times p $ matrix, then $L_{AE} = L_A L_E $
- If $m = n $, then $L_{I_n} = I_{F^n} $
Let $A, B $ and $C $ be matrices s.t. $A(BC)$ is defined. Then $(AB)C $ is also defined and $A(BC) = (AB)C $; that is matrix multiplication is associative.