数学代写|线性代数代写linear algebra代考|MATH1071

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数学代写|线性代数代写linear algebra代考|A Graphical Interpretation of Coordinates

A coordinate system on a set consists of a one-to-one mapping of the points in the set into $\mathbb{R}^n$. For example, ordinary graph paper provides a coordinate system for the plane when one selects perpendicular axes and a unit of measurement on each axis. Figure 1 shows the standard basis $\left{\mathbf{e}_1, \mathbf{e}_2\right}$, the vectors $\mathbf{b}_1\left(=\mathbf{e}_1\right)$ and $\mathbf{b}_2$ from Example 1 , and the vector $\mathbf{x}=\left[\begin{array}{l}1 \ 6\end{array}\right]$. The coordinates 1 and 6 give the location of $\mathbf{x}$ relative to the standard basis: 1 unit in the $\mathbf{e}_1$ direction and 6 units in the $\mathbf{e}_2$ direction.

Figure 2 shows the vectors $\mathbf{b}_1, \mathbf{b}_2$, and $\mathbf{x}$ from Figure 1. (Geometrically, the three vectors lie on a vertical line in both figures.) However, the standard coordinate grid was erased and replaced by a grid especially adapted to the basis $\mathcal{B}$ in Example 1 . The coordinate vector $[\mathbf{x}]_B=\left[\begin{array}{r}-2 \ 3\end{array}\right]$ gives the location of $\mathbf{x}$ on this new coordinate system: $-2$ units in the $\mathbf{b}_1$ direction and 3 units in the $\mathbf{b}_2$ direction.

The coordinates of atoms within the crystal are given relative to the basis for the lattice. For instance,
$$\left[\begin{array}{c} 1 / 2 \ 1 / 2 \ 1 \end{array}\right]$$
identifies the top face-centered atom in the cell in Figure 3(c).
Coordinates in $\mathbb{R}^n$
When a basis $\mathcal{B}$ for $\mathbb{R}^n$ is fixed, the $\mathcal{B}$-coordinate vector of a specified $\mathbf{x}$ is easily found, as in the next example.

数学代写|线性代数代写linear algebra代考|The Coordinate Mapping

Choosing a basis $\mathcal{B}=\left{\mathbf{b}1, \ldots, \mathbf{b}_n\right}$ for a vector space $V$ introduces a coordinate system in $V$. The coordinate mapping $\mathbf{x} \mapsto[\mathbf{x}]{\mathcal{B}}$ connects the possibly unfamiliar space $V$ to the familiar space $\mathbb{R}^n$. See Figure 5. Points in $V$ can now be identified by their new “names.”

It follows that
$$[\mathbf{u}+\mathbf{w}]B=\left[\begin{array}{c} c_1+d_1 \ \vdots \ c_n+d_n \end{array}\right]=\left[\begin{array}{c} c_1 \ \vdots \ c_n \end{array}\right]+\left[\begin{array}{c} d_1 \ \vdots \ d_n \end{array}\right]=[\mathbf{u}]{\mathcal{B}}+[\mathbf{w}]{\mathcal{B}}$$ So the coordinate mapping preserves addition. If $r$ is any scalar, then $$r \mathbf{u}=r\left(c_1 \mathbf{b}_1+\cdots+c_n \mathbf{b}_n\right)=\left(r c_1\right) \mathbf{b}_1+\cdots+\left(r c_n\right) \mathbf{b}_n$$ So $$[r \mathbf{u}]{\mathcal{B}}=\left[\begin{array}{c} r c_1 \ \vdots \ r c_n \end{array}\right]=r\left[\begin{array}{c} c_1 \ \vdots \ c_n \end{array}\right]=r[\mathbf{u}]_{\mathcal{B}}$$
Thus the coordinate mapping also preserves scalar multiplication and hence is a linear transformation. See Exercises 23 and 24 for verification that the coordinate mapping is one-to-one and maps $V$ onto $\mathbb{R}^n$.

The linearity of the coordinate mapping extends to linear combinations, just as in Section 1.8. If $\mathbf{u}1, \ldots, \mathbf{u}_p$ are in $V$ and if $c_1, \ldots, c_p$ are scalars, then $$\left[c_1 \mathbf{u}_1+\cdots+c_p \mathbf{u}_p\right]{\mathcal{B}}=c_1\left[\mathbf{u}1\right]{\mathcal{B}}+\cdots+c_p\left[\mathbf{u}p\right]{\mathcal{B}}$$
In words, (5) says that the $\mathcal{B}$-coordinate vector of a linear combination of $\mathbf{u}_1, \ldots, \mathbf{u}_p$ is the same linear combination of their coordinate vectors.

The coordinate mapping in Theorem 8 is an important example of an isomorphism from $V$ onto $\mathbb{R}^n$. In general, a one-to-one linear transformation from a vector space $V$ onto a vector space $W$ is called an isomorphism from $V$ onto $W$ (iso from the Greek for “the same,” and morph from the Greek for “form” or “structure”). The notation and terminology for $V$ and $W$ may differ, but the two spaces are indistinguishable as vector spaces. Every vector space calculation in $V$ is accurately reproduced in $W$, and vice versa. In particular, any real vector space with a basis of $n$ vectors is indistinguishable from $\mathbb{R}^n$. See Exercises 25 and 26.

线性代数代考

数学代写|线性代数代写linear algebra代考|A Graphical Interpretation of Coordinates

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数学代写|线性代数代写linear algebra代考|The Coordinate Mapping

Imathcal ${B}=\backslash$ left $\left{\backslash m a t h b f{b} 1\right.$, Vdots, $\backslash$ mathbf $\left.{b}_{_} \backslash \backslash r i g h t\right}$

$$[\mathbf{u}+\mathbf{w}] B=\left[c_1+d_1 \vdots c_n+d_n\right]=\left[c_1 \vdots c_n\right]+\left[d_1\right.$$

$$r \mathbf{u}=r\left(c_1 \mathbf{b}1+\cdots+c_n \mathbf{b}_n\right)=\left(r c_1\right) \mathbf{b}_1+\cdots+\left(r c_n\right.$$ 所以 $$[r \mathbf{u}] \mathcal{B}=\left[r c_1 \vdots r c_n\right]=r\left[c_1 \vdots c_n\right]=r[\mathbf{u}]{\mathcal{B}}$$

$$\left[c_1 \mathbf{u}_1+\cdots+c_p \mathbf{u}_p\right] \mathcal{B}=c_1[\mathbf{u} 1] \mathcal{B}+\cdots+c_p[\mathbf{u} p] \mathcal{B}$$

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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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