### 数学代写|几何变换代写transformation geometry代考|МАТН5210

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|几何变换代写transformation geometry代考|The Matrix of a Linear Transformation

We end this chapter on a point of great importance: that every linear transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$ amounts to multiplication by a matrix A. In this case, we say that $\mathbf{A}$ represents $T$ :

Definition 5.1. A linear transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$ is represented by a matrix A when we can compute $T$ using multiplication by $\mathbf{A}$. In other words, A represents $T$ when we have
$$T(\mathbf{x})=\mathbf{A} \mathbf{x}$$
for all inputs $\mathbf{x} \in \mathbf{R}^m$.
As the course proceeds, we’ll learn how to answer almost any question about a linear transformation-like the basic mapping questions listed in Section $3.6$ above – by analyzing the matrix that represents it. We’ll begin acquiring tools for that kind of analysis in Chapter 2. First though, we want to show how to find the matrix that represents a given linear map.
We start with Observation 1.12, which shows how to expand any vector $\mathbf{x}:=\left(x_1, x_2, \ldots, x_n\right) \in \mathbf{R}^m$ as a linear combination of standard basis vectors in a simple way:
$$\left(x_1, x_2, \ldots, x_n\right)=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_m \mathbf{e}_m$$
If we expand a vector $\mathbf{x}$ this way, and then map it into $\mathbf{R}^n$ using a linear transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$, the linearity rules (Definition 4.1) yield \begin{aligned} T(\mathbf{x}) & =T\left(x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_m \mathbf{e}_m\right) \ & =T\left(x_1 \mathbf{e}_1\right)+T\left(x_2 \mathbf{e}_2\right)+\cdots+T\left(x_m \mathbf{e}_m\right) \ & =x_1 T\left(\mathbf{e}_1\right)+x_2 T\left(\mathbf{e}_2\right)+\cdots+x_m T\left(\mathbf{e}_m\right) \end{aligned}
This reveals a powerful fact:

## 数学代写|几何变换代写transformation geometry代考|The Linear System

We now begin to focus on answering the basic mapping questions for linear transformations; that is, for linear mappings
$$T: \mathbf{R}^m \rightarrow \mathbf{R}^n$$
As we observed in Theorem 5.6, every linear transformation is represented by a matrix, via matrix/vector multiplication. Specifically, we have the formula
$$T(\mathbf{x})=\mathbf{A} \mathbf{x}$$
where $\mathbf{A}$ is the matrix whose columns are given by the $T\left(\mathbf{e}_j\right)$ ‘s. For this reason, we can usually reduce questions about the mapping $T$ to calculations involving the matrix $\mathbf{A}$.
In this chapter, we focus on the question of pre-image:
Problem: Given a linear transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$, and a point $\mathbf{b}$ in the range of $T$ how can we find all points in the pre-image $T^{-1}(\mathbf{b})$.

Since every linear map amounts to multiplication by a matrix (and conversely, multiplication by any matrix A defines a linear map), finding $T^{-1}(\mathbf{b})$ is the same as solving $T(\mathbf{x})=\mathbf{b}$ for $\mathbf{x}$. When $T$ is represented by $\mathbf{A}$, we have $T(\mathbf{x})=\mathbf{A} \mathbf{x}$, so the Problem above is exactly the same as this equivalent problem: Given an $n \times m$ matrix $\mathbf{A}$, and a vector $\mathbf{b} \in \mathbf{R}^n$, how can we find every $\mathbf{x} \in \mathbf{R}^m$ that solves the matrix/vector equation
$$\mathbf{A} \mathbf{x}=\mathbf{b}$$
This statement of the problem is nice and terse, but to solve it, we first need to expand its symbols in terms of matrix entries and coordinates. We start with $\mathbf{A}$.

As an $n \times m$ matrix, A has $n$ rows and $m$ columns. Doublesubscripting its entries in the usual way,

# 几何变换代考

## 数学代写|几何变换代写transformation geometry代考|The Matrix of a Linear Transformation

$$T(\mathbf{x})=\mathbf{A x}$$

$$\left(x_1, x_2, \ldots, x_n\right)=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_m \mathbf{e}_m$$

$$T(\mathbf{x})=T\left(x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_m \mathbf{e}_m\right) \quad=T\left(x_1 \mathbf{e}_1\right)+T\left(x_2 \mathbf{e}_2\right)+\cdots+T\left(x_m\right.$$

## 数学代写|几何变换代写transformation geometry代考|The Linear System

$$T: \mathbf{R}^m \rightarrow \mathbf{R}^n$$

$$T(\mathbf{x})=\mathbf{A} \mathbf{x}$$

$$\mathbf{A x}=\mathbf{b}$$

## 有限元方法代写

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## MATLAB代写

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