## 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

## 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|微积分代写Calculus代写|MTH-211

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## 数学代写|微积分代写Calculus代写|ThE Growth OF KinEMatics IN thE WeST

Whereas in dynamics movement is studied in relation to the forces associated with it, in kinematics only its spatial and temporal aspects are considered. It follows that in kinematics we are concerned with the description of movement and not with its causes and it can therefore properly be regarded as the geometry of movement.

If a particle be known to trace a given curve the geometric properties of that curve can be used to predict the subsequent positions of the particle; conversely, if a curve be defined as the path of a point moving under specified conditions, then the laws of kinematics can be utilised to provide information as to certain geometric properties of the curve. For example, knowledge of the instantaneous motion at a given point on the curve enables us to draw the tangent to the curve at that point. The development, in the fourteenth century, of certain important concepts of motion including instantaneous velocity can therefore be seen to have direct and immediate bearing on the study of the tangent properties of curves. Furthermore, the introduction of graphical methods of representation led to the establishment of a link between the velocity-time graph, the total distance covered and the area under the curve, and this in turn is closely connected with the integral calculus (see Figs. 2.8 and 2.9).

The imaginative insights gained by the use of kinematic concepts in geometry were responsible for some of the more powerful methods developed during the seventeenth century for the study of curves. The work of Isaac Barrow, for instance, which certainly influenced Newton, is dominated by the idea of curves generated by moving points and lines.

## 数学代写|微积分代写Calculus代写|The Latitude OF Forms

At Merton College, Oxford, between the years 1328 and 1350, the distinction between kinematics and dynamics was made explicit. In the work of Thomas Bradwardine, William Heytesbury, Richard Swineshead and John Dumbleton the foundations for further study in this field were laid through the clarification and formalisation of a number of important concepts including the notion of instantaneous velocity (velocitas instantanea). ${ }^{\dagger}$

The study of space and motion at Merton College arose from the mediaeval discussion of the intension and remission of forms, i.e. the increase and decrease of the intensity of qualities. The distinction between intension and extension is exemplified in the case of heat by the difference between temperature, or degree of heat, and quantity of heat; in the case of weight between density, or weight per unit volume and total weight. For local motion the distinction is between velocity (or motion) at a given instant (instantaneous velocity) and total motion over a period of time, i.e. distance covered.

## 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|MATH250

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## 数学代写|线性代数代写linear algebra代考|Transformation matrix for non-standard bases

Why use non-standard bases?
Examining a vector in a different basis (axes) may bring out structure related to that basis, which is hidden in the standard representation. It may be a relevant and useful structure. For example, we used to measure the motion of the planets in a basis (axes) with the earth at the centre. Then we discovered that putting the sun at the centre made life simpler – orbits were measured against a basis with the sun at the focus.

For some motions, such as projectiles, our standard basis ( $x y$ axes) may be the most suitable, but for studying other kinds of motions, such as orbits, a polar basis $(r, \theta)$ may work better.

If we use latitudes and longitudes to work out a map then we have been effectively using spherical polar coordinates $(r, \theta, \varphi)$ rather than our standard $x y z$ axes.

Another example is trying to find the forces on an aeroplane as shown in Fig. 5.29. The components parallel and perpendicular to the aeroplane are a lot more useful than the horizontal and vertical components.

In computer games and 3D design software we often want to rotate our $x y z$ axes (basis) to obtain new axes (basis) which are a lot more useful. (See question 7 of Exercises 5.5.)
In crystal structures, we need to use a basis which gives a cleaner set of coordinates called Miller indices. The Miller indices are coordinates used to specify direction and planes in a crystal or lattice. A vector from the origin to the lattice point is normally written in appropriate basis (axes) vectors and then the coordinates are given by the Miller indices.

Many problems in physics can be simplified due to their symmetrical properties if the right basis (axes) is chosen. Choosing a basis (axes) wisely can greatly reduce the amount of arithmetic you have to do.

## 数学代写|线性代数代写linear algebra代考|Composition of linear transformations (mappings)

What do you think the term onto transformation means?
An illustration of an onto transformation is shown in Fig. 5.24.

An onto transformation is when all the information carried over by $T$ fills the whole arrival vector space $W$.
How can we write this in mathematical terms?
Definition (5.18). Let $T: V \rightarrow W$ be a linear transform. The transform $T$ is onto $\Leftrightarrow$ for every $\mathbf{w}$ in the arrival vector space $W$ there exists at least one $\mathbf{v}$ in the start vector space $V$ such that
$$\mathbf{w}=T(\mathbf{v})$$
In other words $T: V \rightarrow W$ is an onto transformation $\Leftrightarrow \operatorname{range}(T)=W$. This means the arriving vectors of $T$ fill all of $W$. We can write this as a proposition:
Proposition (5.19). A linear transformation $T: V \rightarrow W$ is onto $\Leftrightarrow \operatorname{range}(T)=W$.
Proof – Exercises 5.4.

In other mathematical literature, or your lecture notes, you might find the term surjective to mean onto. We will use onto.

Remember that linear transforms are functions and you should be familiar with the concept of a function.
What does composition mean?
Composition means making something by combining parts.
What do you think composition of linear transformation means?
It is the linear transformation created by putting together two or more linear transformations.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Composition of linear transformations (mappings)

$$\mathbf{w}=T(\mathbf{v})$$

## 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。

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Some atoms are unstable and can spontaneously emit mass or radiation. This process is called radioactive decay, and an element whose atoms go spontaneously through this process is called radioactive. Sometimes when an atom emits some of its mass through this process of radioactivity, the remainder of the atom re-forms to make an atom of some new element. For example, radioactive carbon-14 decays into nitrogen; radium, through a number of intermediate radioactive steps, decays into lead.

Experiments have shown that at any given time the rate at which a radioactive element decays (as measured by the number of nuclei that change per unit time) is approximately proportional to the number of radioactive nuclei present. Thus, the decay of a radioactive element is described by the equation $d y / d t=-k y, k>0$. It is conventional to use $-k$, with $k>0$, to emphasize that $y$ is decreasing. If $y_0$ is the number of radioactive nuclei present at time zero, the number still present at any later time $t$ will be
$$y=y_0 e^{-k t}, \quad k>0 .$$
The half-life of a radioactive element is the time expected to pass until half of the radioactive nuclei present in a sample decay. It is an interesting fact that the half-life is a constant that does not depend on the number of radioactive nuclei initially present in the sample, but only on the radioactive substance.

To compute the half-life, let $y_0$ be the number of radioactive nuclei initially present in the sample. Then the number $y$ present at any later time $t$ will be $y=y_0 e^{-k t}$. We seek the value of $t$ at which the number of radioactive nuclei present equals half the original number:
\begin{aligned} y_0 e^{-k t} & =\frac{1}{2} y_0 \ e^{-k t} & =\frac{1}{2} \ -k t & =\ln \frac{1}{2}=-\ln 2 \quad \text { Reciprocal Rule for logarithms } \ t & =\frac{\ln 2}{k} . \end{aligned}

## 数学代写|微积分代写Calculus代写|Heat Transfer: Newton’s Law of Cooling

Hot soup left in a tin cup cools to the temperature of the surrounding air. A hot silver bar immersed in a large tub of water cools to the temperature of the surrounding water. In situations like these, the rate at which an object’s temperature is changing at any given time is roughly proportional to the difference between its temperature and the temperature of the surrounding medium. This observation is called Newton’s Law of Cooling, although it applies to warming as well.

If $H$ is the temperature of the object at time $t$ and $H_S$ is the constant surrounding temperature, then the differential equation is
$$\frac{d H}{d t}=-k\left(H-H_S\right)$$
If we substitute $y$ for $\left(H-H_S\right)$, then
$$\begin{array}{rlrl} \frac{d y}{d t} & =\frac{d}{d t}\left(H-H_S\right)=\frac{d H}{d t}-\frac{d}{d t}\left(H_S\right) & \ & =\frac{d H}{d t}-0 & & \ & =\frac{d H}{d t} & & H_S \text { is a constant. } \ & =-k\left(H-H_S\right) & \ & =-k y . & & \text { Eq. (8) } \ & & H-H_S=y \end{array}$$
We know that the solution of the equation $d y / d t=-k y$ is $y=y_0 e^{-k t}$, where $y(0)=y_0$. Substituting $\left(H-H_S\right)$ for $y$, this says that
$$H-H_S=\left(H_0-H_S\right) e^{-k t},$$
where $H_0$ is the temperature at $t=0$. This equation is the solution to Newton’s Law of Cooling.

# 微积分代考

$$y=y_0 e^{-k t}, \quad k>0 .$$