# Blog · BIGBALLON

## Linear algebra(2)

2017-07-15

• 线性变换(Linear Transformations)， 零空间(Null spaces) 与 值域(ranges)
• 线性变换的矩阵表示(The Matrix Representation of a Linear Transformation)
• 线性变换的组合(Combination of Linear Transformations ) 与 矩阵的乘法(Matrix Multiplication)
• 可逆性(Invertibility) 与 同构(Isomorphisms)
• 坐标变换矩阵(The Change of Coordinate Matrix)

## 2.1 Linear Transformations, null spaces and ranges(线性变换 零空间 与 值域)

• Definition of Linear Transformations:

Let V and W be vector spaces. We call a function 「$T: V \rightarrow W$」 a Linear Transformations(or Linear) from V to W
if for all $x,y \in v$ and $c \in F$, we have (a) $T(x+y) = T(x) + T(y)$ and (b) $T(cx) = cT(x)$.

• Remarks:
• 若 $F=Q$, 则 (a) $\Rightarrow$ (b).
• 一般情况，(a)和(b)是彼此独立的.
• $T(0) = T(0)$.
• $T(x-y)=T(x)-T(y)$.
• $T(\sum_{i=1}^{n}{a_{i}x_{i}})=\sum_{i=1}^{n}{a_{i}T(x_{i})}$.
• $T$ is linear $\Leftrightarrow T(cx+y) = cT(x) + T(y)$ Important
• 「$I: V \rightarrow V$」 is called the identity transformation(相等转换)
• $T_{0}: V \rightarrow V T_{0}(x) = 0$」 is called the zero transformation(零转换)
• Definitions of null spaces and ranges:

「$T: V \rightarrow W$」is Linear.
(i) N(T) = the null space(or kernel) of T. $N(T) = { x \in V : T(x) = 0}$.
(ii) R(T) = the range(or image) of T. $R(T) = { y \in W : \exists x \in V \text{ with } T(x) = y}$.

### Theorem2.1

「$T: V \rightarrow W$」, where V, W are vector spaces and T is linear.
$\Rightarrow$ R(T) and N(T) are subspaces of W and V, respectively.

### Theorem2.2

「$T: V \rightarrow W$」, where V, W are vector spaces and T is linear.
If $\beta = \{ v_{1}, v_{2}, ... v_{n}\}$ if basis for V, then
$R(T) = Span(T(\beta)) = Span(\{ v_{1}, v_{2}, ... v_{n}\})$

### Theorem2.3 (Dimension Thm. or Rank-nullity Thm.)

「$T: V \rightarrow W$」, where V, W are vector spaces and T is linear.
If dim(V) $< \infty$, then nullity(T) + rank(T) = dim(V)
where nullity(T) = dim(N(T)) and rank(T) = dim(R(T)).

### Theorem2.4

「$T: V \rightarrow W$」, where V, W are vector spaces and T is linear.
Then T is one to one. $\Leftrightarrow$ $N(T) = { 0 }$

### Theorem2.5

「$T: V \rightarrow W$」, where V, W are vector spaces and T is linear.
Let dim(V) = dim(W) $< \infty$, then
(i) T is one to one.
(ii) T is onto.
(iii) rank(T) = dim(V) = dim(W)

### Theorem2.6

Let $\{ v_{1},v_{2},...v_{n} \}$ be basis for V. $T(v_{i})=w_{i}, i = 1,2,...,n$,
then $\exists !$ $T: V \rightarrow W$ s.t. $T(v_{i})=w_{i} \text{ for } i = 1,2,...,n$.

## 2.2 The Martix Representation of Linear Transformation (线性变换的矩阵表示)

• Definition of ordered basis:

An ordered basis of V is a finite sequence of linearly independent vectors in V that generates V.

• Definition of the coordinate vector of $x$ relative to $\beta$:

Let $\beta = {u_{1},u_{2},...u_{n}}$ be an ordered basis for V. For $x \in v$, let $a_{1},a_{2},a_{n}$ be the unique scalars s.t. $x = \sum_{i=1}^{n}{a_{i}u_{i}}$.
We define the coordinate vector of $x$ relative to $\beta$, denoted $[x]_{\beta}= \begin{pmatrix} a_{1}\\ a_{2}\\ .\\ .\\ .\\ a_{n}\\ \end{pmatrix}$.

• Remarks:
• $[u_{i}]_{\beta}=e_{i}$.
• $T: V \rightarrow F^{n}$ by $T(x)=[x]_{\beta}$ is a linear trainsformation.
• Definition of the matrix representation(矩阵表示法):：

$\beta = \{ u_{1},u_{2},...u_{n} \}$ and $\gamma = \{ w_{1},w_{2},...w_{m} \}$ be ordered bases for V and W respectively.
Let $T(v_{j})=\sum_{i=1}^{n}{a_{ij}w_{i}} \Rightarrow [T(v_{j})]_{r}= \begin{pmatrix} a_{1j}\\ a_{2j}\\ .\\ .\\ .\\ a_{mj}\\ \end{pmatrix} ,1 \leq j \leq n$ and set $A = (a_{ij})_{m \times n}$.
The matrix $A$ defined above is called the matrix representation of T in the ordered bases $\beta$ and $\gamma$ and write $A=[T]_{\beta}^{\gamma}$.
If $V = W$ and $\beta = \gamma$, then we write $A=[T]_{\beta}$.

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