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Linear algebra(2)

2017-07-15

本章主要内容:


  • 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.
    • $T(0) = T(0)$.
    • $T(x-y)=T(x)-T(y)$.
    • .
    • $T$ is linear $\Leftrightarrow T(cx+y) = cT(x) + T(y)$ Important
    • 「$I: V \rightarrow V$」 is called the identity transformation(相等转换)
    • 」 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 if basis for V, then

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 be basis for V. ,
then $\exists ! $ $T: V \rightarrow W$ s.t. .

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 be an ordered basis for V. For $x \in v$, let be the unique scalars s.t. .
We define the coordinate vector of $x$ relative to $\beta$, denoted .

  • Remarks:
    • .
    • $T: V \rightarrow F^{n}$ by is a linear trainsformation.
  • Definition of the matrix representation(矩阵表示法)::

and be ordered bases for V and W respectively.
Let and set .
The matrix $A$ defined above is called the matrix representation of T in the ordered bases $\beta$ and $\gamma$ and write .
If $V = W$ and $\beta = \gamma$, then we write .

Theorem2.7

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