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Definition: Ordered Basis

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.

Definition: Standard Ordered Basis

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) $.

Example: $\R^2$

$\beta = \set{e_1, e_2}$ is called the standard ordered basis of $\R^2$.

Definition: Coordinate Vector

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

$$[x]_\beta = {\Vcr{a_1}{a_2}{\dots}{a_n}}$$

Example

$\P_2(x)/ \R, \beta = \set{1, x, x^2}$ is an ordered basis.

$f(x) = -5x^2 + x - \frac{1}{2} = -\frac{1}{2} \cdot 1 + 1 \cdot x - 5 \cdot x^2$

$[f(x)]_\beta = {\Vce{- \frac{1}{2}}{1}{-5}}$

Definition: Matrix Representation of Linear Transformations

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 $A = [T]_\beta$.

Example: $T: \R^3 \to \R^2$

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:

$$v \in V, [ T ]_{ \beta }^{ \gamma } [v]_ \beta = [v]_ \gamma $$

Example: $T: \R^3 \to \R^2$

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

$$[ T ]_{ \beta’ }^{ \gamma } = {\Mee{ \mid }{ \mid }{ \mid }{ [ T(\beta’_1) ]_{ \gamma }}{ [ T(\beta’_2) ]_{ \gamma }}{ [ T(\beta’_3) ]_{ \gamma }}{ \mid }{ \mid }{ \mid }} = {\Mwe{ 3 }{ 2 }{ 1 }{ 1 }{ 4 }{ -1 }} $$

Definition: Addition and Scalar Multiplication of Linear Transformations

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:

$$\begin{align*} (T + U)(x) &:= T(x) + U(x) & \forall x \in V \br (cT)(x) &:= cT(x) & \forall c \in F, x \in V \end{align*}$$

Definition: Vector Space of All Linear Transformations From $V$ to $W$

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) $.

Theorem 2.7

Let $V$ and $W$ be vector spaces over $F$. $\L(V, W)$ is a vector space over $F$.

Proof [expand]

exercise. Verify the 8 axioms.

Remarks

Later we will show that given $\dim V = n, \dim W = m , \L(V, W) \cong M_{m \times n}(F)$.

Theorem 2.8

$T: V \to W, U:V \to W$. Let $\beta$ and $\gamma$ be two ordered basis of $V$ and $W$ respectively, then:

  1. $[T+U]^\gamma_\beta = [T]^\gamma_\beta + [U]^\gamma_\beta $
  2. $c \in F, [cT]^\gamma_\beta = c[T]^\gamma_\beta$
Proof [expand]

Let $\dim V = n, \dim W = m$. $\beta = \set{v_1, v_2, …, v_{n}}, \gamma = \set{w_1, w_2, …, w_{m}}$.

$T(v_j) = \sum_{i=1}^{m} a_{ij}w_i$, since $T(v_j) \in R(T) $.

$$[T]^\gamma_\beta = {{\Mqe{{\vcr{ \dots }{ \dots }{ \dots }{ \dots }}}{{\vcr{a_{1j}}{a_{2j}}{a_{3j}}{a_{4j}}}}{{\vcr{ \dots }{ \dots }{ \dots }{ \dots }}}}} = (a_{ij})_{m \times n}$$

Similarly,

$U(v_j) = \sum_{i=1}^{m} b_{ij}w_i$, since $T(v_j) \in R(T) $.

$$[U]^\gamma_\beta = {{\Mqe{{\vcr{ \dots }{ \dots }{ \dots }{ \dots }}}{{\vcr{b_{1j}}{b_{2j}}{b_{3j}}{b_{4j}}}}{{\vcr{ \dots }{ \dots }{ \dots }{ \dots }}}}} = (b_{ij})_{m \times n}$$

$\begin{align*} (T+U)(v_j) &= T(v_j) + U(v_j) \br &= \sum_{i=1}^{m} a_{ij}w_i + \sum_{i=1}^{w} b_{ij}w_i \br &= \sum_{i=1}^{m} (a_{ij} + b_{ij})w_i \end{align*}$

$\therefore$

$\begin{align*} [T+U]^\gamma_\beta &= (a_{ij} + b_{ij})_{m \times n} \br &= [T]^\gamma_\beta + [U]^\gamma_\beta \end{align*}$

$\begin{align*} [cT]^\gamma_\beta &= (ca_{ij})_{m \times n} \br &= c(a_{ij})_{m \times n} \br &= c[T]^\gamma_\beta \end{align*}$