### 数学代写|几何变换代写transformation geometry代考|MATH319

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• 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|Mappings and Transformations

The functions we study in Linear Algebra usually have domains and/or ranges in one of the numeric vector spaces $\mathbf{R}^n$ we introduced in Section 1.

DEFINITION 3.1. A function with numeric vector inputs or outputs is called a mapping or transformation – synonymous terms. A mapping, or transformation is thus simply a function described by a diagram of the form
$$F: \mathbf{R}^n \rightarrow \mathbf{R}^m$$
where $n>1$ and/or $m>1$. Typically, we use uppercase letters like $F$, $G$, or $H$ to label mappings, and from now on, we try to reserve the word function for the case of scalar outputs $(m=1)$.
Example 3.2. A simple mapping
$$J: \mathbf{R}^2 \rightarrow \mathbf{R}^2$$
is given by the rule
$$J(x, y)=(-y, x)$$
This formula makes it easy to compute $J(x, y)$ for any specific input $(x, y) \in \mathbf{R}^2$. For instance, we have
$$J(1,2)=(-2,1), \quad J(-3,5)=(-5,-3), \quad \text { and } \quad J(0,0)=(0,0)$$
Is $J$ one-to-one and/or onto? We leave that as part of Exercise 32 below.

While the domain and range of $J$ are the same, other mappings often have domains and ranges that differ, as the following examples illustrate.
Example $3.3$. The rule
$$F(x, y, z, w)=(x-y, z+w)$$
has four scalar entries in its input, but only two in its output.

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

Recall that both scalar multiplication and matrix/vector multiplication distribute over vector addition (Propositions $1.6$ and 1.25). The definition of linearity generalizes those distributivity rules:

Definition 4.1. A mapping $F: \mathbf{R}^n \rightarrow \mathbf{R}^m$ is linear if it has both these properties:
i) F commutes with vector addition, meaning that for any two inputs $\mathbf{x}, \mathbf{y} \in \mathbf{R}^m$, we have
$$F(\mathbf{x}+\mathbf{y})=F(\mathbf{x})+F(\mathbf{y})$$
ii) $F$ commutes with scalar multiplication, meaning that for any input $\mathbf{x} \in \mathbf{R}^m$ and any scalar $c \in \mathbf{R}$, we have
$$F(c \mathbf{x})=c F(\mathbf{x})$$
Linear mappings are often called linear transformations, and for this reason the favorite symbol for a linear mapping is the letter $T$.
ExAmple $4.2$. The mapping $T: \mathbf{R}^2 \rightarrow \mathbf{R}^2$ given by
(2) $T(a, b)=(2 b, 3 a)$
is linear.
To verify this, we have to show that $T$ has both properties in Definition $4.1$ above.

First property: $T$ commutes with addition: We have to show that for any two vectors $\mathbf{x}=\left(x_1, x_2\right)$, and $\mathbf{y}=\left(y_1, y_2\right)$, we have
(3) $\quad T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$
We do so by expanding each side of the equation separately in coordinates, and checking that they give the same result. On the left, we have
$$T(\mathbf{x}+\mathbf{y})=T\left(\left(\begin{array}{l} x_1 \ x_2 \end{array}\right)+\left(\begin{array}{l} y_1 \ y_2 \end{array}\right)\right)=T\left(x_1+y_1, x_2+y_2\right)$$
and now the rule for $T$, namely (2), reduces this to $T(\mathbf{x}+\mathbf{y})=\left(2\left(x_2+y_2\right), 3\left(x_1+y_1\right)\right)=\left(2 x_2+2 y_2, 3 x_1+3 y_1\right)$

# 几何变换代考

## 数学代写|几何变换代写transformation geometry代考|Mappings and Transformations

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

$$J: \mathbf{R}^2 \rightarrow \mathbf{R}^2$$

$$J(x, y)=(-y, x)$$

$$J(1,2)=(-2,1), \quad J(-3,5)=(-5,-3), \quad \text { and } \quad J(0,0)=(0,0)$$

$$F(x, y, z, w)=(x-y, z+w)$$

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

i) $F$ 通过矢量加法交换，这意味着对于任何两个输入 $\mathbf{x}, \mathbf{y} \in \mathbf{R}^m$ ，我们有
$$F(\mathbf{x}+\mathbf{y})=F(\mathbf{x})+F(\mathbf{y})$$

$$F(c \mathbf{x})=c F(\mathbf{x})$$

(2)给出 $T(a, b)=(2 b, 3 a)$

(3) $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$

$$T(\mathbf{x}+\mathbf{y})=T\left(\left(x_1 x_2\right)+\left(y_1 y_2\right)\right)=T\left(x_1+y_1, x_2+y_2\right)$$

$$T(\mathbf{x}+\mathbf{y})=\left(2\left(x_2+y_2\right), 3\left(x_1+y_1\right)\right)=\left(2 x_2+2 y_2, 3 x_1+3 y_1\right)$$

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