### 数学代写|几何变换代写transformation geometry代考|MATH312

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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|Numeric Vectors

The overarching goal of this book is to impart a sure grasp of the numeric vector functions known as linear transformations. Students will have encountered functions before. We review and expand that familiarity in Section 2 below, and we define linearity in Section 4. Before we can properly discuss these matters though, we must introduce numeric vectors and their basic arithmetic.

DEfinition $1.1$ (Vectors and scalars). A numeric vector (or just vector for short) is an ordered $n$-tuple of the form $\left(x_1, x_2, \ldots, x_n\right)$. Here, each $x_i$-the $i$ th entry (or $i$ th coordinate) of the vector-is a real number.

The $(x, y)$ pairs often used to label points in the plane are familiar examples of vectors with $n=2$, but we allow more than two entries as well. For instance, the triple $(3,-1 / 2,2)$, and the 7-tuple $(1,0,2,0,-2,0,-1)$ are also numeric vectors.
In the linear algebraic setting, we usually call single numbers scalars. This helps highlight the difference between numeric vectors and individual numbers.

Vectors can have many entries, so to clarify and save space, we often label them with single bold letters instead of writing out all their entries. For example, we might define
\begin{aligned} \mathbf{x} & :=\left(x_1, x_2, \ldots, x_n\right) \ \mathbf{a} & :=\left(a_1, a_2, a_3, a_4\right) \ \mathbf{b} & :=(-5,0,1) \end{aligned}
and then use $\mathbf{x}$, a, or $\mathbf{b}$ to indicate the associated vector. We use boldface to distinguish vectors from scalars. For instance, the same letters, without boldface, would typically represent scalars, as in $x=5$, $a=-4.2$, or $b=\pi$.
Often, we write numeric vectors vertically instead of horizontally, in which case $\mathbf{x}, \mathbf{a}$, and $\mathbf{b}$ above would look like this:

$$\mathbf{x}=\left(\begin{array}{r} x_1 \ x_2 \ \vdots \ x_m \end{array}\right), \quad \mathbf{a}=\left(\begin{array}{c} a_1 \ a_2 \ a_3 \ a_4 \end{array}\right), \quad \mathbf{b}=\left(\begin{array}{r} -5 \ 0 \ 1 \end{array}\right)$$
In our approach to the subject (unlike some others) we draw absolutely no distinction between
$$\left(x_1, x_2, \ldots, x_n\right) \text { and }\left(\begin{array}{r} x_1 \ x_2 \ \vdots \ x_n \end{array}\right)$$
These are merely different notations for the same vector – the very same mathematical object.

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

Now that we’re familiar with numeric vectors and matrices, we can consider vector functions – functions that take numeric vectors as inputs and produce them as outputs. The ultimate goal of this book is to give students a detailed understanding of linear vector functions, both algebraically, and geometrically. Here and in Section 3, we lay out the basic vocabulary for the kinds of questions one seeks to answer for any vector function, linear or not. Then, in Section 4, we introduce linearity, and with these building blocks all in place, we can at least state the main questions we’ll be answering in later chapters.
2.1. Domain, image, and range. Roughly speaking, a function is an input-output rule. Here is is a more precise formal definition.
DEFINITION 2.2. A function is an input/output relation specified by three data:
i) A domain set $X$ containing all allowed inputs,
ii) A range set $Y$ containing all allowed outputs, and
iii) A rule $f$ that assigns exactly one output $f(x)$ to every input $x$ in the domain.

We typically signal all three of these at once with a simple diagram like this:
$$f: X \rightarrow Y$$
For instance, if we apply the rule $T(x, y)=x+y$ to any input pair $(x, y) \in \mathbf{R}^2$, we get a scalar output in $\mathbf{R}$, and we can summarize this situation by writing $T: \mathbf{R}^2 \rightarrow \mathbf{R}$.

Technically, function and mapping are synonyms, but we will soon reserve the term function for the situation where (as with $T$ above) the range is just $\mathbf{R}$. When the range is $\mathbf{R}^n$ for some $n>1$, we typically prefer the term mapping or transformation.

# 几何变换代考

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

$$\mathbf{x}:=\left(x_1, x_2, \ldots, x_n\right) \mathbf{a} \quad:=\left(a_1, a_2, a_3, a_4\right) \mathbf{b}:=(-5,0,1)$$

$$\mathbf{x}=\left(x_1 x_2 \vdots x_m\right), \quad \mathbf{a}=\left(a_1 a_2 a_3 a_4\right), \quad \mathbf{b}=\left(\begin{array}{lll} -5 & 0 & 1 \end{array}\right)$$

$$\left(x_1, x_2, \ldots, x_n\right) \text { and }\left(x_1 x_2 \vdots x_n\right)$$

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

2.1. 域 图像和范围。粗略地说，一个函数就是一个输入输出规则。这是一个更精确的正式定 义

i) 域集 $X$ 包含所有允许的输入，
ii) 范围集 $Y$ 包含所有允许的输出，以及
iii) 规则 $f$ 恰好分配一个输出 $f(x)$ 对每个输入 $x$ 在域中。

$$f: X \rightarrow Y$$

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## MATLAB代写

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