# 线性代数网课代修|数值线性代数代写Numerical linear algebra代考

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|A Short Review of Linear Algebra

1. The sets of natural numbers, integers, rational numbers, real numbers, and complex numbers are denoted by $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$, respectively.
2. We use the “colon equal” symbol $v:=e$ to indicate that the symbol $v$ is defined by the expression $e$.
3. $\mathbb{R}^{n}$ is the set of $n$-tuples of real numbers which we will represent as bold face column vectors. Thus $\boldsymbol{x} \in \mathbb{R}^{n}$ means $$\boldsymbol{x}=\left[\begin{array}{c} x_{1} \ x_{2} \ \vdots \ x_{n} \end{array}\right],$$ where $x_{i} \in \mathbb{R}$ for $i=1, \ldots, n$. Row vectors are normally identified using the transpose operation. Thus if $x \in \mathbb{R}^{n}$ then $x$ is a column vector and $x^{T}$ is a row vector.

## 线性代数作业代写linear algebra代考|Vector Spaces and Subspaces

Many mathematical systems have analogous properties to vectors in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. Definition $1.1$ (Real Vector Space) A real vector space is a nonempty set $\mathcal{V}$, whose objects are called vectors, together with two operations $+: \mathcal{V} \times \mathcal{V} \longrightarrow \mathcal{V}$ and $\cdot: \mathbb{R} \times \mathcal{V} \longrightarrow \mathcal{V}$, called addition and scalar multiplication, satisfying the following axioms for all vectors $u, v, w$ in $\mathcal{V}$ and scalars $c, d$ in $\mathbb{R}$. (V1) The sum $u+v$ is in $\mathcal{V}$, (V2) $u+v=v+u$, (V3) $u+(v+w)=(u+v)+w$, (V4) There is a zero vector 0 such that $u+0=u$, (V5) For each $u$ in $\mathcal{V}$ there is a vector $-u$ in $\mathcal{V}$ such that $u+(-u)=0$, (S1) The scalar multiple $c \cdot u$ is in $\mathcal{V}$, (S2) $c \cdot(u+v)=c \cdot u+c \cdot v$, (S3) $(c+d) \cdot u=c \cdot u+d \cdot u$, (S4) $c \cdot(d \cdot u)=(c d) \cdot u$, (S5) $1 \cdot u=u$. The scalar multiplication symbol – is often omitted, writing $c v$ instead of $c \cdot v$. We define $u-v:=u+(-v)$. We call $\mathcal{V}$ a complex vector space if the scalars consist of all complex numbers $\mathbb{C}$. In this book a vector space is either real or complex. From the axioms it follows that

1. The zero vector is unique.
2. For each $u \in \mathcal{V}$ the negative $-u$ of $u$ is unique.
3. $0 u=0, c 0=0$, and $-u=(-1) u$. Here are some examples
4. The spaces $\mathbb{R}^{n}$ and $\mathbb{C}^{n}$, where $n \in \mathbb{N}$, are real and complex vector spaces, respectively.
5. Let $\mathcal{D}$ be a subset of $\mathbb{R}$ and $d \in \mathbb{N}$. The set $\mathcal{V}$ of all functions $f, g: \mathcal{D} \rightarrow \mathbb{R}^{d}$ is a real vector space with $$(f+g)(t):=f(t)+g(t), \quad(c f)(t):=c f(t), \quad t \in \mathcal{D}, \quad c \in \mathbb{R} .$$

## 线性代数作业代写linear algebra代考|Linear Systems

Consider a linear system $a_{11} x_{1}+a_{12} x_{2}+\cdots+a_{1 n} x_{n}=b_{1}$ $a_{21} x_{1}+a_{22} x_{2}+\cdots+a_{2 n} x_{n}=b_{2}$ $\vdots \vdots \vdots \vdots \vdots$ of $m$ equations in $n$ unknowns. Here for all $i, j$, the coefficients $a_{i j}$, the unknowns $x_{j}$, and the components $b_{i}$ of the right hand side are real or complex numbers. The system can be written as a vector equation $$x_{1} a_{1}+x_{2} a_{2}+\cdots+x_{n} a_{n}=b,$$

## 线性代数作业代写linear algebra代考 $\mid \mathrm{A}$ Short Review of Linear Algebra

㧴们使用“冒号等号”符号 $v:=e$ 表示该符号 $v$ 由表达式定义 $e$.

$\mathbb{R}^{n}$ 是集合 $n$-我们将表示为粗体列向量的实数元组。因此 $\boldsymbol{x} \in \mathbb{R}^{n}$ 方法 $$\boldsymbol{x}=\left[\begin{array}{lll} & & \ x_{1} & x_{2} & x_{n} \end{array}\right]$$ 在哪里 $x_{i} \in \mathbb{R}$ 为了 $i=1, \ldots, n$. 行向量通常使用转置操作来识别。因此，如果 $x \in \mathbb{R}^{n}$ 然后 $x$ 是一个列向量并且 $x^{T}$ 是一个行向量。

## 线性代数作业代写linear algebra代 考|Vector Spaces and Subspaces

$0 u=0, c 0=0, \quad$ 和 $-u=(-1) u$. 这里有些例子

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions