# 数学代写|线性代数代写linear algebra代考|МАTH1051

## 数学代写|线性代数代写linear algebra代考|Linear Transformations

Definition 43 Let $U$ and $V$ be vector spaces over $\mathbb{K}$. A function $T: U \rightarrow V$ is called a linear transformation if, for all $\boldsymbol{x}, \boldsymbol{y} \in U$ and $\alpha \in \mathbb{K}$,
\begin{aligned} T(\boldsymbol{x}+\boldsymbol{y}) &=T(\boldsymbol{x})+T(\boldsymbol{y}), \ T(\alpha \boldsymbol{x}) &=\alpha T(\boldsymbol{x}) . \end{aligned}
In other words, a function $T: U \rightarrow V$ is a linear transformation if it is additive (5.1) and homogeneous (5.2). As usual, $U$ is the domain of $T$ and $V$ is its codomain.

Proposition 5.1 Let $U$ and $V$ be vector spaces over $\mathbb{K}$ and let $T: U \rightarrow V$ be a linear transformation. Then
$$T\left(\mathbf{0}_U\right)=\mathbf{0}_V,$$
where $\mathbf{0}_U$ is the zero vector in $U$ and $\mathbf{0}_V$ is the zero vector in $V$.
Proof Let $\boldsymbol{x}$ be a vector in U. Hence, by (5.1),
$$T(\boldsymbol{x})=T\left(\boldsymbol{x}+\mathbf{0}_U\right)=T(\boldsymbol{x})+T\left(\mathbf{0}_U\right),$$
from which follows that $T\left(\mathbf{0}_U\right)=\mathbf{0}_V$.
Example 5.1 Find which of the following functions are linear transformations.
a) $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a reflection relative to the $x$-axis.
b) $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ is an orthogonal projection on the xy-plane.
c) $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a translation by the vector $\boldsymbol{u}=(1,0)$.
The function $T$ in a) is defined, for all $\boldsymbol{x}=\left(x_1, x_2\right)$ in $\mathbb{R}^2$, by
$$T\left(x_1, x_2\right)=\left(x_1,-x_2\right) .$$

## 数学代写|线性代数代写linear algebra代考|Matrix Representations

Let $T: \mathbb{K}^n \rightarrow \mathbb{K}^k$ be a linear transformation and let $\mathcal{E}n=\left(e_1, e_2, \ldots, e_n\right)$ be the ordered standard basis of $\mathbb{K}^n$ and $\mathcal{E}_k$ the ordered standard basis of $\mathbb{K}^k$. Then, for $\boldsymbol{x}=\left(x_1, x_2, \ldots, x_n\right)$, we have $$T(\boldsymbol{x})=T\left(x_1 \boldsymbol{e}_1+x_2 \boldsymbol{e}_2+\cdots+x_n \boldsymbol{e}_n\right)=x_1 T\left(\boldsymbol{e}_1\right)+x_2 T\left(\boldsymbol{e}_2\right)+\cdots+x_n T\left(\boldsymbol{e}_n\right)$$ Denoting by $[\boldsymbol{x}]$ the vector column version of a vector $\boldsymbol{x}$, Observe that, given a vector $\boldsymbol{u} \in \mathbb{R}^m$, we have that $[\boldsymbol{u}]=[\boldsymbol{u}]{\mathcal{E}m}$, where $[\boldsymbol{u}]{\mathcal{E}m}$ is the coordinate vector of $\boldsymbol{u}$ relative to the basis $\mathcal{E}_m$. Hence, we can now rewrite $(5.4)$ as $$[T(\boldsymbol{x})]{\mathcal{E}k}=\underbrace{\left[\left[T\left(\boldsymbol{e}_1\right)\right]{\mathcal{E}k} \mid\left[\left[T\left(\boldsymbol{e}_2\right)\right]{\mathcal{E}k}|\ldots|\left[\ldots\left(\boldsymbol{e}_n\right)\right]{\mathcal{E}k}\right]\right.}{[T]{\mathcal{E}_k, \mathcal{E}_n}}[\boldsymbol{x}]{\mathcal{E}n}$$ That is, $$[T(\boldsymbol{x})]{\mathcal{E}k}=[T]{\mathcal{E}k, \mathcal{E}_n}[\boldsymbol{x}]{\mathcal{E}n},$$ where $[T]{\mathcal{E}_k, \mathcal{E}_n}$ is a $k \times n$ matrix called the matrix of $T$ relative to the standard bases of the domain $\mathbb{K}^n$ and codomain $\mathbb{K}^k$. In what follows this matrix might be denoted simply by $[T]$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Linear Transformations

$$T(\boldsymbol{x}+\boldsymbol{y})=T(\boldsymbol{x})+T(\boldsymbol{y}), T(\alpha \boldsymbol{x}) \quad=\alpha T(\boldsymbol{x}) .$$

$$T\left(\mathbf{0}_U\right)=\mathbf{0}_V,$$

$$T(\boldsymbol{x})=T\left(\boldsymbol{x}+\mathbf{0}_U\right)=T(\boldsymbol{x})+T\left(\mathbf{0}_U\right),$$

b) $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ 是 xy 平面上的正交投影。
C) $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ 是向量的平移 $\boldsymbol{u}=(1,0)$.

$$T\left(x_1, x_2\right)=\left(x_1,-x_2\right) .$$

## 数学代写|线性代数代写linear algebra代考|Matrix Representations

$$T(\boldsymbol{x})=T\left(x_1 \boldsymbol{e}_1+x_2 \boldsymbol{e}_2+\cdots+x_n \boldsymbol{e}_n\right)=x_1 T\left(\boldsymbol{e}_1\right)+x_2 T\left(\boldsymbol{e}_2\right)+\cdots+x_n T\left(\boldsymbol{e}_n\right)$$

$$[T(\boldsymbol{x})] \mathcal{E} k=\underbrace{\left[\left[T\left(\boldsymbol{e}_1\right)\right] \mathcal{E} k \mid\left[\left[T\left(\boldsymbol{e}_2\right)\right] \mathcal{E} k|\ldots|\left[\ldots\left(\boldsymbol{e}_n\right)\right] \mathcal{E} k\right]\right.}[T] \mathcal{E}_k, \mathcal{E}_n[\boldsymbol{x}] \mathcal{E} n$$

$$[T(\boldsymbol{x})] \mathcal{E} k=[T] \mathcal{E} k, \mathcal{E}_n[\boldsymbol{x}] \mathcal{E} n,$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: