Let $V$ be a finite-dimensional vector space. An ordered basis for $V$ is a basis for $V$ with a specific preassigned order on the vectors.
For the vector space $F^n $, we call $\set{ e_1, e_2, \dots, e_{ n }} $ the standard ordered basis for $F^n $. Similarly, for the vector space $P_n(F) $, we call $\set{ 1, x, \dots, x^n } $ the standard ordered basis for $P_n(F) $.
$\beta = \set{e_1, e_2}$ is called the standard ordered basis of $\R^2$.
Let $\beta = \set{u_1, u_2, \dots, u_{n}}$ be an ordered basis of a finite-dimensional vector space, $V$ is a vector space over a field $F$. For $x \in V$, exists unique scalars $a_1, a_2, \dots, a_{n} \in F$ s.t. $x= a_1u_1 + a_2u_2 + \dots + a_{n} u_{n}$.
We define the coordinate vector of $x $ relative to $\beta $, denoted $[x]_ \beta $, by
$$[x]_\beta = {\Vcr{a_1}{a_2}{\dots}{a_n}}$$
$\P_2(x)/ \R, \beta = \set{1, x, x^2}$ is an ordered basis.
$f(x) = -5x^2 + x - \frac{1}{2} = -\frac{1}{2} \cdot 1 + 1 \cdot x - 5 \cdot x^2$
$[f(x)]_\beta = {\Vce{- \frac{1}{2}}{1}{-5}}$
Let $T: V \to W, \dim V = n, \dim W = m$. Let $\beta = \set{v_1, v_2, \dots, v_{n}}, \gamma = \set{w_1, w_2, \dots, w_{m}}$ be ordered basis of $V$ and $W$ respectively.
Now choose $x_j \in \beta$ and consider $T(v_j)$. Since $\gamma$ is a basis of $W , \exists a_{1j}, a_{2j}, \dots, a_{mj} \in F$ s.t. $T(v_j) = a_{1j}w_1 + a_{2j}w_2 + \dots + a_{mj}w_{m}= \sum_{i=1}^{m} a_{ij}w_i$.
$$[T(v_j)]_\gamma = {\Vcr{ a_{1j}}{ a_{2j}}{ \dots }{ a_{mj}}}$$
Let $A$ be a matrix s.t. Its columns are the coordinates vector $[T(v_1)]_\gamma, [T(v_2)]_\gamma, \ldots, [T(v_n)]_\gamma$.
$$A = \Mqr{\vce{\mid}{[T(v_1)]_\gamma}{\mid}}{\vce{\mid}{[T(v_2)]_\gamma}{\mid}}{\dots}{\vce{\mid}{[T(v_n)]_\gamma}{\mid}}$$
This matrix $A$ is called the matrix representation of $T$ in the ordered basis $\beta$ and $\gamma$ and write $A = [T]^\gamma_\beta$.
In particular, if $V = W$ and $\beta = \gamma$, then we write $A = [T]_\beta$.
Find the matrix representation of $T(x, y, z) = (2x-y, x+z)$.
Let $\beta = \set{e_1, e_2, e_3}, \gamma = \set{e_1', e_2'}$.
$T(e_1) = (2, 1) = 2e_1' +1 e_2'$
$T(e_2) = (-1, 0) = -1e_1' + 0e_2'$
$T(e_3) = (0, 1) = 0e_1' + 1e_2'$
$\therefore $ the matrix representation of $T $ in $\beta $ and $\gamma $ is
$$[ T ]_{ \beta }^{ \gamma } = {\Mee{ \mid }{ \mid }{ \mid }{ [ T(e_1) ]_{ \gamma }}{ [ T(e_2) ]_{ \gamma }}{ [ T(e_3) ]_{ \gamma }}{ \mid }{ \mid }{ \mid }} = {\Mwe{ 2 }{ -1 }{ 0 }{ 1 }{ 0 }{ 1 }} $$
with which we have:
$$v \in V, [ T ]_{ \beta }^{ \gamma } [v]_ \beta = [v]_ \gamma $$
Find the matrix representation of $T(x, y, z) = (2x-y, x+z)$.
Let $\beta' = \set{(1,-1, 0), (1,0, 3), (1,1,-2)}, \gamma = \set{(1, 0), (0, 1)}$.
$T(\beta_1') = (3,1) = 3e_1' + 1e_2'$
$T(\beta_2') = (2,4) = 2e_1' + 4e_2'$
$T(\beta_3') = (1,-1) = 1e_1' + -1e_2'$
$\therefore $ the matrix representation of $T $ in $\beta' $ and $\gamma $ is
$$[ T ]_{ \beta' }^{ \gamma } = {\Mee{ \mid }{ \mid }{ \mid }{ [ T(\beta'_1) ]_{ \gamma }}{ [ T(\beta'_2) ]_{ \gamma }}{ [ T(\beta'_3) ]_{ \gamma }}{ \mid }{ \mid }{ \mid }} = {\Mwe{ 3 }{ 2 }{ 1 }{ 1 }{ 4 }{ -1 }} $$
Let $T: V \to W$ and $U: V \to W$ be two linear transformations, where V and W be vector spaces over a field $F $, we then define:
$$\begin{align*} (T + U)(x) &:= T(x) + U(x) & \forall x \in V \br (cT)(x) &:= cT(x) & \forall c \in F, x \in V \end{align*}$$
Let $V$ and $W$ be two vector spaces over $F$. We denote the vector space of all linear transformations from $V$ into $W$ by $\L (V, W)$.
In particular, if $V = W $, we simply write $\L(V) $ instead of $\L(V, W) $.
Let $V$ and $W$ be vector spaces over $F$. $\L(V, W)$ is a vector space over $F$.
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Later we will show that given $\dim V = n, \dim W = m , \L(V, W) \cong M_{m \times n}(F)$.
$T: V \to W, U:V \to W$. Let $\beta$ and $\gamma$ be two ordered basis of $V$ and $W$ respectively, then:
- $[T+U]^\gamma_\beta = [T]^\gamma_\beta + [U]^\gamma_\beta $
- $c \in F, [cT]^\gamma_\beta = c[T]^\gamma_\beta$
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