### 数学代写|线性代数代写linear algebra代考|MATH1071

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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|线性代数代写linear algebra代考|Solution Spaces

In this section, we consider solution sets of linear equations. If an equation has $n$ variables, its solution set is a subset of $\mathbb{R}^{n}$. When is this set a vector space?
Example 2.4.9 Let $V \subseteq \mathbb{R}^{3}$ be the set of all solutions to the equation $x_{1}+3 x_{2}-x_{3}=0$. That is,
$$V=\left{\left(\begin{array}{l} x_{1} \ x_{2} \ x_{3} \end{array}\right) \in \mathbb{R}^{3} \mid x_{1}+3 x_{2}-x_{3}=0\right}$$
The set $V$ together with the operations $+$ and – inherited from $\mathbb{R}^{3}$ forms a vector space.
Proof. Let $V$ be the set defined above and let
$$u=\left(\begin{array}{l} u_{1} \ u_{2} \ u_{3} \end{array}\right), v=\left(\begin{array}{l} v_{1} \ v_{2} \ v_{3} \end{array}\right) \in V .$$
Let $\alpha \in \mathbb{R}$. Notice that Properties (P3)-(P7) and (P10) of Definition $2.3 .5$ only depend on the definition of addition and scalar multiplication on $\mathbb{R}^{3}$ and, therefore, are inherited properties from $\mathbb{R}^{3}$. Hence we need only check properties (P1), (P2), (P8), and (P9). Since $u, v \in V$, we know that

$$u_{1}+3 u_{2}-u_{3}=0 \text { and } v_{1}+3 v_{2}-v_{3}=0$$
We will use this result to show the closure properties.
First, notice that $u+v=\left(\begin{array}{l}u_{1}+v_{1} \ u_{2}+v_{2} \ u_{3}+v_{3}\end{array}\right)$. Notice also that
\begin{aligned} \left(u_{1}+v_{1}\right)+3\left(u_{2}+v_{2}\right)-\left(u_{3}+v_{3}\right) &=\left(u_{1}+3 u_{2}-u_{3}\right)+\left(v_{1}+3 v_{2}-v_{3}\right) \ &=0+0=0 \end{aligned}
Thus, $u+v \in V$. Since $u$ and $v$ are arbitrary vectors in $V$ it follows that $V$ is closed under addition.
Next, notice that $\alpha \cdot u=\left(\begin{array}{l}\alpha u_{1} \ \alpha u_{2} \ \alpha u_{3}\end{array}\right)$ and
\begin{aligned} \alpha u_{1}+3 \alpha u_{2}-\alpha u_{3} &=\alpha\left(u_{1}+3 u_{2}-u_{3}\right) \ &=\alpha(0)=0 \end{aligned}
Therefore, $\alpha u \in V$. Hence $V$ is closed under scalar multiplication.

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

In many areas of mathematics, we learn about concepts that relate to vector spaces, though the details of vector space properties may be simply assumed or not well established. In this section, we look at some of these concepts and recognize how vector space properties are present.
Sequence Spaces
Here, we explore sequences such as those discussed in Calculus. We consider the set of all sequences in the context of vector space properties. First, we give a formal definition of sequences.

A sequence of real numbers is a function $s: \mathbb{N} \rightarrow \mathbb{R}$. That is, $s(n)=a_{n}$ for $n=1,2, \cdots$ where $a_{n} \in \mathbb{R}$. A sequence is denoted $\left{a_{n}\right}$. Let $\mathcal{S}(\mathbb{R})$ be the set of all sequences. Let $\left{a_{n}\right}$ and $\left{b_{n}\right}$ be sequences in $\mathcal{S}(\mathbb{R})$ and $\alpha$ in $\mathbb{R}$. Define sequence addition and scalar multiplication with a sequence by
$$\left{a_{n}\right}+\left{b_{n}\right}=\left{a_{n}+b_{n}\right} \text { and } \alpha \cdot\left{a_{n}\right}=\left{\alpha a_{n}\right} .$$
In Exercise 15, we show that $\mathcal{S}(\mathbb{R})$, with these (element-wise) operations, forms a vector space over $\mathbb{R}$.

Example 2.4.15 (Eventually Zero Sequences) Let $\mathcal{S}{\text {fin }}(\mathbb{R})$ be the set of all sequences that have a finite number of nonzero terms. Then $\mathcal{S}{\text {fin }}(\mathbb{R})$ is a vector space with operations as defined in Definition 2.4.14. (See Exercise 16.)

We find vector space properties for sequences to be very useful in the development of calculus concepts such as limits. For example, if we want to apply a limit to the sum of sequences, we need to know that the sum of two sequences is indeed a sequence. More of these concepts will be discussed later, after developing more linear algebra ideas.

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

PetPics, a pet photography company specializing in portraits, wants to post photos for clients to review, but to protect their artistic work, they only post electronic versions that have copyright text. The text is

added to all images, produced by the company, by overwriting, with zeros, in the appropriate pixels, as shown in Figure 2.18. Only pictures that have zeros in these pixels are considered legitimate images.
The company also wants to allow clients to make some adjustments to the pictures: the adjustments include brightening/darkening, and adding background or little figures like hearts, flowers, or squirrels. It turns out that these operations can all be accomplished by adding other legitimate images and multiplying by scalars, as defined in Section $2.3$.

It is certainly true that the set of all legitimate images of the company’s standard $(m \times n)$-pixel size is contained in the vector space $\mathcal{I}_{m \times n}$ of all $m \times n$ images, so we could mathematically work in this larger space. But, astute employees of the company who enjoy thinking about linear algebra notice that actually the set of legitimate images itself satisfies the 10 properties of a vector space. Specifically, adding any two images with the copyright text (for example, adding a squirrel to a portrait of a golden retriever) produces another image with the same copyright text, and multiplying an image with the copyright text by a scalar (say, to brighten it) still results in an image with the copyright text. Hence, it suffices to work with the smaller set of legitimate images.

In fact, very often the sets of objects that we want to focus on are actually only subsets of larger vector spaces, and it is useful to know when such a set forms a vector space separately from the larger vector space.
Here are some examples of subsets of vector spaces that we have encountered so far.

1. Solution sets of homogeneous linear equations, with $n$ variables, are subsets of $\mathbb{R}^{n}$.
2. Radiographs are images with nonnegative values and represent a subset of the larger vector space of images with the given geometry.
3. The set of even functions on $\mathbb{R}$ is a subset of the vector space of functions on $\mathbb{R}$.
4. Polynomials of order 3 form a subset of the vector space $\mathcal{P}_{5}(\mathbb{R})$.
5. Heat states on a rod in a diffusion welding process (the collection of which is $H_{m}(\mathbb{R})$ ) form a subset of all possible heat states because the temperature is fixed at the ends of the rod.
6. The set of sequences with exactly 10 nonzero terms is a subset of the set of sequences with a finite number of terms.

Even though operations like vector addition and scalar multiplication on the subset are typically the same as the operations on the larger parent spaces, we still often wish to work in the smaller more relevant subset rather than thinking about the larger ambient space. When does the subset behave like a vector space in its own right? In general, when is a subset of a vector space also a vector space?

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

V=\left{\left(\begin{array}{l} x_{1} \ x_{2} \ x_{3} \end{array}\right) \in \mathbb{R}^{3} \中间 x_{1}+3 x_{2}-x_{3}=0\right}V=\left{\left(\begin{array}{l} x_{1} \ x_{2} \ x_{3} \end{array}\right) \in \mathbb{R}^{3} \中间 x_{1}+3 x_{2}-x_{3}=0\right}

(在1+在1)+3(在2+在2)−(在3+在3)=(在1+3在2−在3)+(在1+3在2−在3) =0+0=0

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

\left{a_{n}\right}+\left{b_{n}\right}=\left{a_{n}+b_{n}\right} \text { and } \alpha \cdot\left{a_ {n}\right}=\left{\alpha a_{n}\right} 。\left{a_{n}\right}+\left{b_{n}\right}=\left{a_{n}+b_{n}\right} \text { and } \alpha \cdot\left{a_ {n}\right}=\left{\alpha a_{n}\right} 。

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

PetPics是一家专门从事人像摄影的宠物摄影公司，希望发布照片供客户审阅，但为了保护他们的艺术作品，他们只发布具有版权文本的电子版本。文字是

1. 齐次线性方程组的解集，其中n变量，是的子集Rn.
2. 射线照片是具有非负值的图像，表示具有给定几何形状的图像的较大矢量空间的子集。
3. 偶函数集R是函数向量空间的子集R.
4. 3 阶多项式形成向量空间的子集磷5(R).
5. 扩散焊接过程中棒上的热态（其集合是H米(R)) 形成所有可能的热状态的子集，因为温度固定在棒的末端。
6. 恰好有 10 个非零项的序列集是具有有限项数的序列集的子集。

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