面向工程師的矩陣代數

Matrix Algebra for Engineers

This course is all about matrices, and concisely covers the linear algebra that an engineer should know.

香港科技大學

Coursera

數學與統計

簡單(初級)

13 小時

課程概況

This course is all about matrices, and concisely covers the linear algebra that an engineer should know. We define matrices and how to add and multiply them, and introduce some special types of matrices. We describe the Gaussian elimination algorithm used to solve systems of linear equations and the corresponding LU decomposition of a matrix. We explain the concept of vector spaces and define the main vocabulary of linear algebra. We develop the theory of determinants and use it to solve the eigenvalue problem.

After each video, there are problems to solve and I have tried to choose problems that exemplify the main idea of the lecture. I try to give enough problems for students to solidify their understanding of the material, but not so many that students feel overwhelmed and drop out. I do encourage students to attempt the given problems, but if they get stuck, full solutions can be found in the lecture notes for the course.

The mathematics in this matrix algebra course is presented at the level of an advanced high school student, but typically students would take this course after completing a university-level single variable calculus course. There are no derivatives or integrals in this course, but student’s are expected to have a certain level of mathematical maturity. Nevertheless, anyone who wants to learn the basics of matrix algebra is welcome to join.

Lecture notes may be downloaded at
https://bookboon.com/en/matrix-algebra-for-engineers-ebook
or
http://www.math.ust.hk/~machas/matrix-algebra-for-engineers.pdf

Watch the course overview video at
youtu.be/IZcyZHomFQc

課程大綱

周1
完成時間為 7 小時
MATRICES
In this week's lectures, we learn about matrices. Matrices are rectangular arrays of numbers or other mathematical objects and are fundamental to engineering mathematics. We will define matrices and how to add and multiply them, discuss some special matrices such as the identity and zero matrix, learn about transposes and inverses, and define orthogonal and permutation matrices.
11 個視頻 (總計 84 分鐘), 25 個閱讀材料, 5 個測驗

周2
完成時間為 3 小時
SYSTEMS OF LINEAR EQUATIONS
In this week's lectures, we learn about solving a system of linear equations. A system of linear equations can be written in matrix form, and we can solve using Gaussian elimination. We will learn how to bring a matrix to reduced row echelon form, and how this can be used to compute a matrix inverse. We will also learn how to find the LU decomposition of a matrix, and how to use this decomposition to efficiently solve a system of linear equations.
7 個視頻 (總計 71 分鐘), 6 個閱讀材料, 3 個測驗

周3
完成時間為 6 小時
VECTOR SPACES
In this week's lectures, we learn about vector spaces. A vector space consists of a set of vectors and a set of scalars that is closed under vector addition and scalar multiplication and that satisfies the usual rules of arithmetic. We will learn some of the vocabulary and phrases of linear algebra, such as linear independence, span, basis and dimension. We will learn about the four fundamental subspaces of a matrix, the Gram-Schmidt process, orthogonal projection, and the matrix formulation of the least-squares problem of drawing a straight line to fit noisy data.
13 個視頻 (總計 140 分鐘), 14 個閱讀材料, 5 個測驗

周4
完成時間為 6 小時
EIGENVALUES AND EIGENVECTORS
In this week's lectures, we will learn about determinants and the eigenvalue problem. We will learn how to compute determinants using a Laplace expansion, the Leibniz formula, or by row or column elimination. We will formulate the eigenvalue problem and learn how to find the eigenvalues and eigenvectors of a matrix. We will learn how to diagonalize a matrix using its eigenvalues and eigenvectors, and how this leads to an easy calculation of a matrix raised to a power.

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