## 数学代写|线性代数代写linear algebra代考|MTH-230

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## 数学代写|线性代数代写linear algebra代考|Direct Sum Decompositions

Throughout this section, $V$ will be a vector space over a field $F$, and $W_i$, for $i=1, \ldots, k$, will be subspaces of $V$. For facts and general reading for this section, see [HK71].

Definitions:
The sum of subspaces $W_i$, for $i=1, \ldots, k$, is $\sum_{i=1}^k W_i=W_1+\cdots+W_k=\left{\mathbf{w}1+\cdots+\mathbf{w}_k \mid \mathbf{w}_i \in W_i\right}$. The sum $W_1+\cdots+W_k$ is a direct sum if for all $i=1, \ldots, k$, we have $W_i \cap \sum{j \neq i} W_j={0}$. $W=W_1 \oplus \cdots \oplus W_k$ denotes that $W=W_1+\cdots+W_k$ and the sum is direct. The subspaces $W_i$, for $i=i, \ldots, k$, are independent if for $\mathbf{w}_i \in W_i, \mathbf{w}_1+\cdots+\mathbf{w}_k=\mathbf{0}$ implies $\mathbf{w}_i=\mathbf{0}$ for all $i=1, \ldots, k$. Let $V_i$, for $i=1, \ldots, k$, be vector spaces over $F$. The external direct sum of the $V_i$, denoted $V_1 \times \cdots \times V_k$, is the cartesian product of $V_i$, for $i=1, \ldots, k$, with coordinate-wise operations. Let $W$ be a subspace of $V$. An additive coset of $W$ is a subset of the form $v+W={v+w \mid w \in W}$ with $v \in V$. The quotient of $V$ by $W$, denoted $V / W$, is the set of additive cosets of $W$ with operations $\left(v_1+W\right)+\left(v_2+W\right)=\left(v_1+v_2\right)+W$ and $c(v+W)=(c v)+W$, for any $c \in F$. Let $V=W \oplus U$, let $\mathcal{B}_W$ and $\mathcal{B}_U$ be bases for $W$ and $U$ respectively, and let $\mathcal{B}=\mathcal{B}_W \cup \mathcal{B}_U$. The induced basis of $\mathcal{B}$ in $V / W$ is the set of vectors $\left{u+W \mid u \in \mathcal{B}_U\right}$.

Facts:

1. $W=W_1 \oplus W_2$ if and only if $W=W_1+W_2$ and $W_1 \cap W_2={0}$.
2. If $W$ is a subspace of $V$, then there exists a subspace $U$ of $V$ such that $V=W \oplus U$. Note that $U$ is not usually unique.
3. Let $W=W_1+\cdots+W_k$. The following are equivalent:
• $W=W_1 \oplus \cdots \oplus W_k$. That is, for all $i=1, \ldots, k$, we have $W_i \cap \sum_{j \neq i} W_j={0}$.
• $W_i \cap \sum_{j=1}^{i-1} W_j={0}$, for all $i=2, \ldots, k$.
• For each $\mathbf{w} \in W, \mathbf{w}$ can be expressed in exactly one way as a sum of vectors in $W_1, \ldots, W_k$. That is, there exist unique $\mathbf{w}_i \in W_i$, such that $\mathbf{w}=\mathbf{w}_1+\cdots+\mathbf{w}_k$.
• The subspaces $W_i$, for $i=1, \ldots, k$, are independent.
• If $\mathcal{B}i$ is an (ordered) basis for $W_i$, then $\mathcal{B}=\bigcup{i=1}^k \mathcal{B}_i$ is an (ordered) basis for $W$.
1. If $\mathcal{B}$ is a basis for $V$ and $\mathcal{B}$ is partitioned into disjoint subsets $\mathcal{B}_i$, for $i=1, \ldots, k$, then $V=\operatorname{Span}\left(\mathcal{B}_1\right) \oplus \cdots \oplus \operatorname{Span}\left(\mathcal{B}_k\right)$.
2. If $S$ is a linearly independent subset of $V$ and $S$ is partitioned into disjoint subsets $S_i$, for $i=1, \ldots, k$, then the subspaces $\operatorname{Span}\left(S_1\right), \ldots, \operatorname{Span}\left(S_k\right)$ are independent.
3. If $V$ is finite dimensional and $V=W_1+\cdots+W_k$, then $\operatorname{dim}(V)=\operatorname{dim}\left(W_1\right)+\cdots+\operatorname{dim}\left(W_k\right)$ if and only if $V=W_1 \oplus \cdots \oplus W_k$.
4. Let $V_i$, for $i=1, \ldots, k$, be vector spaces over $F$.
• $V_1 \times \cdots \times V_k$ is a vector space over $F$.
• $\widehat{V}_i=\left{\left(0, \ldots, 0, v_i, 0, \ldots, 0\right) \mid v_i \in V_i\right}$ (where $v_i$ is the $i$ th coordinate) is a subspace of $V_1 \times \cdots \times V_k$.
• $V_1 \times \cdots \times V_k=\widehat{V}_1 \oplus \cdots \oplus \widehat{V}_k$.
• If $V_i$, for $i=1, \ldots, k$, are finite dimensional, then $\operatorname{dim} \widehat{V}_i=\operatorname{dim} V_i$ and $\operatorname{dim}\left(V_1 \times \cdots \times V_k\right)=$ $\operatorname{dim} V_1+\cdots+\operatorname{dim} V_k$.
1. If $W$ is a subspace of $V$, then the quotient $V / W$ is a vector space over $F$.
2. Let $V=W \oplus U$, let $\mathcal{B}_W$ and $\mathcal{B}_U$ be bases for $W$ and $U$ respectively, and let $\mathcal{B}=\mathcal{B}_W \cup \mathcal{B}_U$. The induced basis of $\mathcal{B}$ in $V / W$ is a basis for $V / W$ and $\operatorname{dim}(V / W)=\operatorname{dim} U$.

## 数学代写|线性代数代写linear algebra代考|Matrix Range, Null Space, Rank, and the Dimension Theorem

Definitions:
For any matrix $A \in F^{m \times n}$, the range of $A$, denoted by range $(A)$, is the set of all linear combinations of the columns of $A$. If $A=\left[\mathbf{m}_1 \mathbf{m}_2 \ldots \mathbf{m}_n\right]$, then $\operatorname{range}(A)=\operatorname{Span}\left(\mathbf{m}_1, \mathbf{m}_2, \ldots, \mathbf{m}_n\right)$. The $\operatorname{range}$ of $A$ is also called the column space of $A$.

The row space of $A$, denoted by $\operatorname{RS}(A)$, is the set of all linear combinations of the rows of $A$. If $A=\left[\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_m\right]^T$, then $\operatorname{RS}(A)=\operatorname{Span}\left(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_m\right)$.

The kernel of $A$, denoted by $\operatorname{ker}(A)$, is the set of all solutions to the homogeneous equation $A \mathbf{x}=\mathbf{0}$. The kernel of $A$ is also called the null space of $A$, and its dimension is called the nullity of $A$, denoted by $\operatorname{null}(A)$.

The rank of $A$, denoted by $\operatorname{rank}(A)$, is the number of leading entries in the reduced row echelon form of $A$ (or any row echelon form of $A$ ). (See Section 1.3 for more information.)

$A, B \in F^{m \times n}$ are equivalent if $B=C_1^{-1} A C_2$ for some invertible matrices $C_1 \in F^{m \times m}$ and $C_2 \in F^{n \times n}$. $A, B \in F^{n \times n}$ are similar if $B=C^{-1} A C$ for some invertible matrix $C \in F^{n \times n}$. For square matrices $A_1 \in F^{n_1 \times n_1}, \ldots, A_k \in F^{n_k \times n_k}$, the matrix direct sum $A=A_1 \oplus \cdots \oplus A_k$ is the block diagonal matrix with the matrices $A_i$ down the diagonal. That is, $A=\left[\begin{array}{ccc}A_1 & & \ & \ddots & \ & & \ \mathbf{0} & & A_k\end{array}\right]$, where $A \in F^{n \times n}$ with $n=\sum_{i=1}^k n_i$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Direct Sum Decompositions

$W=W_1 \oplus W_2$ 当且仅当$W=W_1+W_2$和$W_1 \cap W_2={0}$。

$W=W_1 \oplus \cdots \oplus W_k$． 也就是说，对于所有$i=1, \ldots, k$，我们有$W_i \cap \sum_{j \neq i} W_j={0}$。

$W_i \cap \sum_{j=1}^{i-1} W_j={0}$，为所有$i=2, \ldots, k$。

$V_1 \times \cdots \times V_k$ 是$F$上的向量空间。

$\widehat{V}_i=\left{\left(0, \ldots, 0, v_i, 0, \ldots, 0\right) \mid v_i \in V_i\right}$ (其中$v_i$是$i$的第一个坐标)是$V_1 \times \cdots \times V_k$的子空间。

$V_1 \times \cdots \times V_k=\widehat{V}_1 \oplus \cdots \oplus \widehat{V}_k$．

## 数学代写|线性代数代写linear algebra代考|Basis and Dimension of a Vector Space

$\mathcal{B}$ 是一个线性无关的集合，那么

$\operatorname{Span}(\mathcal{B})=V$．

$F^n$的标准基是$F^n$的基，因此$\operatorname{dim} F^n=n$也是如此。

$S$ 是$V$的基础。
$S$横跨$V$。
$S$是线性无关的。

[Lay03, Section 4.4]如果$\mathcal{B}=\left{\mathbf{b}_1, \ldots, \mathbf{b}_p\right}$是一个向量空间$V$的基，那么每个$\mathbf{x} \in V$可以表示为$\mathcal{B}$中向量的唯一线性组合。也就是说，对于每个$\mathbf{x} \in V$，都有一组唯一的标量$c_1, c_2, \ldots, c_p$，使得$\mathbf{x}=c_1 \mathbf{b}_1+c_2 \mathbf{b}_2+\cdots+c_p \mathbf{b}_p$。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|线性代数代写linear algebra代考|MTH204

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## 数学代写|线性代数代写linear algebra代考|Matrix Inverses and Elementary Matrices

Invertibility is a strong and useful property. For example, when a linear system $A \mathbf{x}=\mathbf{b}$ has an invertible coefficient matrix $A$, it has a unique solution. The various characterizations of invertibility in Fact 10 below are also quite useful. Throughout this section, $F$ will denote a field.

Definitions:
An $n \times n$ matrix $A$ is invertible, or nonsingular, if there exists another $n \times n$ matrix $B$, called the inverse of $A$, such that $A B=B A=I_n$. The inverse of $A$ is denoted $A^{-1}$ (cf. Fact 1). If no such $B$ exists, $A$ is not invertible, or singular.

For an $n \times n$ matrix and a positive integer $m$, the $\boldsymbol{m}$ th power of $A$ is $A^m=\underbrace{A A \ldots A}_{m \text { copies of } A}$. It is also convenient to define $A^0=I_n$. If $A$ is invertible, then $A^{-m}=\left(A^{-1}\right)^m$.

An elementary matrix is a square matrix obtained by doing one elementary row operation to an identity matrix. Thus, there are three types:

1. A multiple of one row of $I_n$ has been added to a different row.
2. Two different rows of $I_n$ have been exchanged.
3. One row of $I_n$ has been multiplied by a nonzero scalar.

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

This section discusses the $L U$ and $P L U$ factorizations of a matrix that arise naturally when Gaussian Elimination is done. Several other factorizations are widely used for real and complex matrices, such as the QR, Singular Value, and Cholesky Factorizations. (See Chapter 5 and Chapter 38.) Throughout this section, $F$ will denote a field and $A$ will denote a matrix over $F$. The material in this section and additional background can be found in [GV96, Sec. 3.2].
Definitions:
Let $A$ be a matrix of any shape.
An $L U$ factorization, or triangular factorization, of $A$ is a factorization $A=L U$ where $L$ is a square unit lower triangular matrix and $U$ is upper triangular. A PLU factorization of $A$ is a factorization of the form $P A=L U$ where $P$ is a permutation matrix, $L$ is square unit lower triangular, and $U$ is upper triangular. An $L D U$ factorization of $A$ is a factorization $A=L D U$ where $L$ is a square unit lower triangular matrix, $D$ is a square diagonal matrix, and $U$ is a unit upper triangular matrix.

A $P L D U$ factorization of $A$ is a factorization $P A=L D U$ where $P$ is a permutation matrix, $L$ is a square unit lower triangular matrix, $D$ is a square diagonal matrix, and $U$ is a unit upper triangular matrix.
Facts: [GV96, Sec. 3.2]

1. Let $A$ be square. If each leading principal submatrix of $A$, except possibly $A$ itself, is invertible, then $A$ has an $L U$ factorization. When $A$ is invertible, $A$ has an $L U$ factorization if and only if each leading principal submatrix of $A$ is invertible; in this case, the $L U$ factorization is unique and there is also a unique $L D U$ factorization of $A$.
2. Any matrix $A$ has a PLU factorization. Algorithm 1 (Section 1.3) performs the addition of multiples of pivot rows to lower rows and perhaps row exchanges to obtain an REF matrix $U$. If instead, the same series of row exchanges are done to $A$ before any pivoting, this creates $P A$ where $P$ is a permutation matrix, and then $P A$ can be reduced to $U$ without row exchanges. That is, there exist unit lower triangular matrices $E_j$ such that $E_k \ldots E_1(P A)=U$. It follows that $P A=L U$, where $L=\left(E_k \ldots E_1\right)^{-1}$ is unit lower triangular and $U$ is upper triangular.
3. In most professional software packages, the standard method for solving a square linear system $A \mathbf{x}=\mathbf{b}$, for which $A$ is invertible, is to reduce $A$ to an REF matrix $U$ as in Fact 2 above, choosing row exchanges by a strategy to reduce pivot size. By keeping track of the exchanges and pivot operations done, this produces a $P L U$ factorization of $A$. Then $A=P^T L U$ and $P^T L U \mathbf{x}=\mathbf{b}$ is the equation to be solved. Using forward substitution, $P^{\mathrm{T}} L \mathbf{y}=\mathbf{b}$ can be solved quickly for $\mathbf{y}$, and then $U \mathbf{x}=\mathbf{y}$ can either be solved quickly for $\mathbf{x}$ by back substitutution, or be seen to be inconsistent. This method gives accurate results for most problems. There are other types of solution methods that can work more accurately or efficiently for special types of matrices.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Matrix Inverses and Elementary Matrices

\mathbf{u} \cdot \mathbf{v}=\mathbf{u}^T \mathbf{v}=u_1 v_1+u_2 v_2+u_3 v_3+\cdots+u_n v_n
$$Remember, the answer was a scalar not a vector. This inner product was named the dot product (also called the scalar product) in \mathbb{R}^n. This is the usual (or standard) inner product in \mathbb{R}^n but there are many other types of inner products in \mathbb{R}^n. For the general vector space, the inner product is denoted by \langle\mathbf{u}, \mathbf{v}\rangle rather than \mathbf{u} \cdot \mathbf{v}. For the general vector space, the definition of inner product is based on Proposition (2.6) of chapter 2 and is given by: Definition (4.1). An inner product on a real vector space V is an operation which assigns to each pair of vectors, \mathbf{u} and \mathbf{v}, a unique real number \langle\mathbf{u}, \mathbf{v}\rangle which satisfies the following axioms for all vectors \mathbf{u}, \mathbf{v} and \mathbf{w} in V and all scalars k. (i) \langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle \quad [commutative law] (ii) \langle\mathbf{u}+\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{w}\rangle+\langle\mathbf{v}, \mathbf{w}\rangle \quad [distributive law] (iii) \langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle \quad [taking out the scalar k ] (iv) \langle\mathbf{u}, \mathbf{u}\rangle \geq 0 and we have \langle\mathbf{u}, \mathbf{u}\rangle=0 \Leftrightarrow \mathbf{u}=\mathbf{O} [Means the inner product between the same vectors is zero or positive.] A real vector space which satisfies these axioms is called a real inner product space. Note that evaluating \langle,\rangle gives a real number (scalar) not a vector. Next we give some examples of inner product spaces. ## 数学代写|线性代数代写linear algebra代考|Properties of inner products Proposition (4.2). Let \mathbf{u}, \mathbf{v} and \mathbf{w} be vectors in a real inner product space V and k be any real scalar. We have the following properties of inner products: (i) \langle\mathbf{u}, \mathbf{O}\rangle=\langle\mathbf{O}, \mathbf{v}\rangle=0 (ii) \langle\mathbf{u}, k \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle (iii) \langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle How do we prove these properties? We use the axioms of inner products stated in Definition (4.1). Proof of (i). We can write the zero vector as 0(\mathbf{O}) because 0(\mathbf{O})=\mathbf{O}. Using the axioms of definition (4.1) we have$$
\begin{aligned}
\langle\mathbf{u}, \mathbf{O}\rangle & =\langle\mathbf{u}, 0(\mathbf{O})\rangle & & \
& =\langle 0(\mathbf{O}), \mathbf{u}\rangle & & {[\text { by part (i) of (4.1) which is }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \
& =0\langle\mathbf{O}, \mathbf{u}\rangle & & {[\text { by part (iii) of (4.1) which is }\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle] } \
& =0 & &
\end{aligned}
$$Similarly \langle\mathbf{O}, \mathbf{v}\rangle=0. Proof of (ii). The inner product is commutative, \langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle, which means we can switch the vectors around. We have$$
\begin{aligned}
\langle\mathbf{u}, k \mathbf{v}\rangle & =\langle k \mathbf{v}, \mathbf{u}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \
& =k\langle\mathbf{v}, \mathbf{u}\rangle & & {[\text { by part (iii) of (4.1) which is }\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle] } \
& =k\langle\mathbf{u}, \mathbf{v}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] }
\end{aligned}
$$Proof of (iii). We have$$
\begin{aligned}
\langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle & =\langle\mathbf{v}+\mathbf{w}, \mathbf{u}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \
& =\langle\mathbf{v}, \mathbf{u}\rangle+\langle\mathbf{w}, \mathbf{u}\rangle & & {\left[\begin{array}{c}
\text { by part (ii) of }(4.1) \text { which is } \
\langle\mathbf{v}+\mathbf{w}, \mathbf{u}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle+\langle\mathbf{w}, \mathbf{u}\rangle
\end{array}\right] } \
& =\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] }
\end{aligned}
$$# 线性代数代考 ## 数学代写|线性代数代写linear algebra代考|Definition of inner product 在第二章中我们是如何定义内积或点积的? 设\mathbf{u}=\left(\begin{array}{c}u_1 \ \vdots \ u_n\end{array}\right)和\mathbf{v}=\left(\begin{array}{c}v_1 \ \vdots \ v_n\end{array}\right)为\mathbb{R}^n中的向量，则\mathbf{u}与\mathbf{v}的内积表示为\mathbf{u} \cdot \mathbf{v}$$
\mathbf{u} \cdot \mathbf{v}=\mathbf{u}^T \mathbf{v}=u_1 v_1+u_2 v_2+u_3 v_3+\cdots+u_n v_n
$$记住，答案是一个标量而不是一个向量。这个内积在\mathbb{R}^n中被称为点积(也称为标量积)。这是\mathbb{R}^n中常见的(或标准的)内积，但\mathbb{R}^n中还有许多其他类型的内积。 对于一般向量空间，内积用\langle\mathbf{u}, \mathbf{v}\rangle而不是\mathbf{u} \cdot \mathbf{v}表示。对于一般向量空间，内积的定义基于第2章的命题(2.6)，由式给出: 定义(4.1)。实向量空间V上的内积是一个运算，它将一个唯一的实数\langle\mathbf{u}, \mathbf{v}\rangle赋给每一对向量\mathbf{u}和\mathbf{v}，该实数满足以下公理，适用于V中的所有向量\mathbf{u}, \mathbf{v}和\mathbf{w}以及所有标量k。 (i) \langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle \quad[交换律] (ii) \langle\mathbf{u}+\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{w}\rangle+\langle\mathbf{v}, \mathbf{w}\rangle \quad[分配律] (iii) \langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle \quad[取出标量k] (iv) \langle\mathbf{u}, \mathbf{u}\rangle \geq 0，我们有\langle\mathbf{u}, \mathbf{u}\rangle=0 \Leftrightarrow \mathbf{u}=\mathbf{O}[意味着相同向量之间的内积为零或正。] 满足这些公理的实向量空间称为实内积空间。注意计算内积空间的\langle, \rangle gives a real number (scalar) not a vector. Next we give some examples。 ## 数学代写|线性代数代写linear algebra代考|Properties of inner products 命题(4.2)。设\mathbf{u}, \mathbf{v}和\mathbf{w}是实内积空间中的向量V和k是任意实标量。我们有内积的下列性质: (i) \langle\mathbf{u}, \mathbf{O}\rangle=\langle\mathbf{O}, \mathbf{v}\rangle=0 (ii) \langle\mathbf{u}, k \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle (三)\langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle 我们如何证明这些性质? 我们使用定义(4.1)中所述的内积公理。 证明(i)。 我们可以把零向量写成0(\mathbf{O})因为0(\mathbf{O})=\mathbf{O}。使用定义公理(4.1)，我们有$$
\begin{aligned}
\langle\mathbf{u}, \mathbf{O}\rangle & =\langle\mathbf{u}, 0(\mathbf{O})\rangle & & \
& =\langle 0(\mathbf{O}), \mathbf{u}\rangle & & {[\text { by part (i) of (4.1) which is }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \
& =0\langle\mathbf{O}, \mathbf{u}\rangle & & {[\text { by part (iii) of (4.1) which is }\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle] } \
& =0 & &
\end{aligned}
$$类似的\langle\mathbf{O}, \mathbf{v}\rangle=0。 证明(ii)。 内积是可交换的，\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle，这意味着我们可以交换向量。我们有$$
\begin{aligned}
\langle\mathbf{u}, k \mathbf{v}\rangle & =\langle k \mathbf{v}, \mathbf{u}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \
& =k\langle\mathbf{v}, \mathbf{u}\rangle & & {[\text { by part (iii) of (4.1) which is }\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle] } \
& =k\langle\mathbf{u}, \mathbf{v}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] }
\end{aligned}
$$证明(iii)。 我们有$$
\begin{aligned}
\langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle & =\langle\mathbf{v}+\mathbf{w}, \mathbf{u}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \
& =\langle\mathbf{v}, \mathbf{u}\rangle+\langle\mathbf{w}, \mathbf{u}\rangle & & {\left[\begin{array}{c}
\text { by part (ii) of }(4.1) \text { which is } \
\langle\mathbf{v}+\mathbf{w}, \mathbf{u}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle+\langle\mathbf{w}, \mathbf{u}\rangle
\end{array}\right] } \
& =\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] }
\end{aligned}


## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。