### 数学代写|线性代数代写linear algebra代考|MAST10007

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|线性代数代写linear algebra代考|The Geometry of Systems of Equations

It turns out that there is an intimate connection between solutions to systems of equations in two variables and the geometry of lines in $\mathbb{R}^{2}$. We recall the graphical method to solving systems below. Although you will likely have already done this in previous classes, we include it here so that you can put this knowledge into the context of solution sets to systems of equations as classified in Theorem 2.2.20.
We begin with the following simple example:
Example 2.2.27 Let us consider $u=\left(\begin{array}{c}2 \ -3\end{array}\right), v=\left(\begin{array}{l}1 \ 1\end{array}\right)$, and $w=\left(\begin{array}{l}2 \ 3\end{array}\right) \in \mathbb{R}^{2}$. Suppose we want to know if we can express $u$ using arithmetic operations on $v$ and $w$. In other words, we want to know if there are scalars $x, y$ so that
$$\left(\begin{array}{c} 2 \ -3 \end{array}\right)=x \cdot\left(\begin{array}{l} 1 \ 1 \end{array}\right)+y \cdot\left(\begin{array}{l} 2 \ 3 \end{array}\right) \text {. }$$
We can rewrite the right-hand side of the vector equation so that we have the equation with two vectors
$$\left(\begin{array}{c} 2 \ -3 \end{array}\right)=\left(\begin{array}{l} x+2 y \ x+3 y \end{array}\right) .$$
The equivalent system of linear equations with 2 equations and 2 variables is
\begin{aligned} &x+2 y=2 \ &x+3 y=-3 \end{aligned}
Equations (2.18) and (2.19) are equations of lines in $\mathbb{R}^{2}$, that is, the set of pairs $(x, y)$ that satisfy each equation is the set of points on each respective line. Hence, finding $x$ and $y$ that satisfy both equations amounts to finding all points $(x, y)$ that are on both lines. If we graph these two lines, we can see that they appear to cross at one point, $(12,-5)$, and nowhere else, so we estimate $x=12$ and $y=-5$ is the only solution of the two equations. (See Figure 2.9.) You can also algebraically verify that $(12,5)$ is a solution to the system.

## 数学代写|线性代数代写linear algebra代考|Images and Image Arithmetic

In Section $2.1$ we saw that if you add two images, you get a new image, and that if you multiply an image by a scalar, you get a new image. We represented a rectangular pixelated image as an array of values, or equivalently, as a rectangular array of grayscale patches. This is a very natural idea in the context of digital photography.

Recall the definition of an image given in Section 2.1. We repeat it here, and follow the definition by some examples of images with different geometric arrangements.

An image is a finite ordered list of real values with an associated geometric arrangement.
Four examples of arrays along with an index system specifying the order of patches can be seen in Figure 2.11. As an image, each patch would also have a numerical value indicating the brightness of the patch (not shown in the figure). The first is a regular pixel array commonly used for digital photography. The second is a hexagonal pattern which also nicely tiles a plane. The third is a map of the African continent and Madagascar subdivided by country. The fourth is a square pixel set with enhanced resolution toward the center of the field of interest. It should be clear from the definition that images are not matrices. Only the first example might be confused with a matrix.

We first fix a particular geometric arrangement of pixels (and let $n$ denote the number of pixels in the arrangement). Then an image is precisely described by its (ordered) intensity values. With this determined, we formalize the notions of scalar multiplication and addition on images that were developed in the previous section.

## 数学代写|线性代数代写linear algebra代考|Vectors and Vector Spaces

In the last section, we saw that the set of $4 \times 4$ images, together with real scalars, satisfies several natural properties. There are in fact many other sets of objects that also have these properties.

One class of objects with these properties are the vectors that you may have seen in a course in multivariable calculus or physics. In those courses, vectors are objects with a fixed number, say $n$, of values put together into an ordered tuple. That is, the word vector may bring to mind something that looks like $\langle a, b\rangle,\langle a, b, c\rangle$, or $\left\langle a_{1}, a_{2}, \ldots, a_{n}\right\rangle$. Maybe you have even seen notation like any of the following:
$$(a, b), \quad(a, b, c), \quad\left(a_{1}, a_{2}, \ldots, a_{n}\right),\left(\begin{array}{l} a \ b \ c \end{array}\right), \quad\left[\begin{array}{c} a \ b \ c \end{array}\right],\left(\begin{array}{c} a_{1} \ a_{2} \ \vdots \ a_{n} \end{array}\right),\left[\begin{array}{c} a_{1} \ a_{2} \ \vdots \ a_{n} \end{array}\right]$$
called vectors as well.
In this section, we generalize the notion of a vector. In particular, we will understand that images and other classes of objects can be vectors in an appropriate context. When we consider objects like brain images, radiographs, or heat state signatures, it is often useful to understand them as collections having certain natural mathematical properties. Indeed, we will develop mathematical tools that can be used on all such sets, and these tools will be instrumental in accomplishing our application tasks.
We haven’t yet made the definition of a vector space (or even a vector) rigorous. We still have some more setup to do. In this text, we will primarily use two scalar fields ${ }^{6}: \mathbb{R}$ and $Z_{2}$. The field $Z_{2}$ is the two element (or binary) set ${0,1}$ with addition and multiplication defined modulo 2 . That is, addition defined modulo 2, means that:

$$0+0=0, \quad 0+1=1+0=1, \quad \text { and } 1+1=0$$
And, multiplication defined modulo 2 means
$$0 \cdot 0=0, \quad 0 \cdot 1=1 \cdot 0=0, \quad \text { and } 1 \cdot 1=1 .$$
We can think of the two elements as “on” and “off” and the operations as binary operations. If we add 1 , we flip the switch and if we add 0 , we do nothing. We know that $Z_{2}$ is closed under scalar multiplication and vector addition.

## 数学代写|线性代数代写linear algebra代考|The Geometry of Systems of Equations

(2 −3)=X⋅(1 1)+是⋅(2 3).

(2 −3)=(X+2是 X+3是).

X+2是=2 X+3是=−3

## 数学代写|线性代数代写linear algebra代考|Vectors and Vector Spaces

(一个,b),(一个,b,C),(一个1,一个2,…,一个n),(一个 b C),[一个 b C],(一个1 一个2 ⋮ 一个n),[一个1 一个2 ⋮ 一个n]

0+0=0,0+1=1+0=1, 和 1+1=0

0⋅0=0,0⋅1=1⋅0=0, 和 1⋅1=1.

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