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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

## 计算机代写|机器学习代写machine learning代考|Linear Transformation as Matrix Multiplication

A common question that beginners have is why we need vectors and matrices and what we can do with them. We can easily spot that vectors may be viewed as special matrices. However, it must be noted that vectors and matrices represent very different concepts in mathematics. An $n$ dimensional vector can be viewed as a point in an $n$-dimensional space if we interpret each number in the vector as the coordinate along an axis. Each axis in turn can be viewed as some measurement of one particular characteristic of an object. In other words, vectors can be viewed as an abstract way to represent objects in mathematics. On the other hand, a matrix represents a motion of all points in a space (i.e., one particular way to move any point in a space into a different position in another space). Alternatively, a matrix can be viewed as a particular way to transform the representations of objects from one space to another. More importantly, the exact algorithm to implement such motion is to take advantage of a matrix operation, called matrix multiplication, which is defined as shown in Figure 2.1.

We denote this as $\mathbf{y}=\mathbf{A x}$ for short. Using the matrix multiplication, any point $\mathbf{x}$ in the first space $\mathbb{R}^n$ is transformed into another point $\mathbf{y}$ in a different space $\mathbb{R}^m$. The exact mapping between $\mathbf{x}$ and $\mathbf{y}$ depends on all numbers in the matrix $\mathbf{A}$. If $\mathbf{A}$ is a square matrix in $\mathbb{R}^{n \times n}$, this mapping can also be viewed as transforming one point $\mathbf{x} \in \mathbb{R}^n$ into another point $\mathbf{y}$ in the same space $\mathbb{R}^n$.

However, this matrix multiplication cannot implement any arbitrary mapping between two spaces. The matrix multiplication actually can only implement a small subset of all possible mappings called linear transformations. As shown in Figure 2.2, a linear transformation is a mapping from the first space $\mathbb{R}^n$ to another space $\mathbb{R}^m$ that must satisfy two conditions: (i) the origin in $\mathbb{R}^n$ is mapped to the origin in $\mathbb{R}^m$; (ii) every straight line in $\mathbb{R}^n$ is always mapped to a straight line (or a single point) in $\mathbb{R}^m$. Other mappings that do not satisfy these two conditions are called nonlinear transformations, which must be implemented by other methods rather than matrix multiplication.

## 计算机代写|机器学习代写machine learning代考|Basic Matrix Operations

For any square matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$, we can compute a real number for it, called the determinant, denoted as $|\mathbf{A}|(\in \mathbb{R})$. As we know, a square matrix A represents a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^n$, and it will transform any unit hypercube in the original space into a polyhedron in the new space. The determinant $|\mathbf{A}|$ represents the volume of the polyhedron in the new space.

We often use I to represent a special square matrix, called an identity matrix, that has all $1 \mathrm{~s}$ in its diagonal and 0 s everywhere else. For a square matrix $\mathbf{A}$, if we can find another square matrix, denoted as $\mathbf{A}^{-1}$, that satisfies
$$\mathbf{A}^{-1} \mathbf{A}=\mathbf{A A}^{-1}=\mathbf{I},$$
we call $\mathbf{A}^{-1}$ the inverse matrix of $\mathbf{A}$. We say $\mathbf{A}$ is invertible if its inverse matrix $\mathbf{A}^{-1}$ exists.

The inner product between any two $n$-dimensional vectors (e.g., $\mathbf{w} \in \mathbb{R}^n$ and $\mathbf{x} \in \mathbb{R}^n$ ) is defined as the sum of all element-wise multiplications between them, denoted as $\mathbf{w} \cdot \mathbf{x}(\in \mathbb{R})$. We can further represent the inner product using the matrix transpose and multiplication as follows:
$$\mathbf{w} \cdot \mathbf{x} \triangleq \sum_{i=1}^n w_i x_i=\mathbf{w}^{\top} \mathbf{x}=\mathbf{x}^{\top} \mathbf{w} .$$
The norm of a vector $\mathbf{w}$ (a.k.a. the $L_2$ norm), denoted as $|\mathbf{w}|$, is defined as the square root of the inner product with itself. The meaning of the norm $|\mathbf{w}|$ represents the length of the vector $\mathbf{w}$ in the Euclidean space:
$$|\mathbf{w}|^2=\mathbf{w} \cdot \mathbf{w}=\sum_{i=1}^n w_i^2=\mathbf{w}^{\top} \mathbf{w} .$$

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Basic Matrix Operations

$$\mathbf{A}^{-1} \mathbf{A}=\mathbf{A A}^{-1}=\mathbf{I},$$

$$\mathbf{w} \cdot \mathbf{x} \triangleq \sum_{i=1}^n w_i x_i=\mathbf{w}^{\top} \mathbf{x}=\mathbf{x}^{\top} \mathbf{w} .$$

$$|\mathbf{w}|^2=\mathbf{w} \cdot \mathbf{w}=\sum_{i=1}^n w_i^2=\mathbf{w}^{\top} \mathbf{w} .$$

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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。

assignmentutor™您的专属作业导师
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