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指定教材
《线性代数导论》(Introduction to Linear Algebra)第三版(2003年三月)，Gilbert Strang着，Wellesley-Cambridge Press出版。
线性代数课程目标 e
本课程(18.06)的目的为使学生了解矩阵的性质并应用。
下列为其重要演算及其所本之观念：
- 以消去法解方阵 Ax = b 。(包括高斯消去法、乘数消去法、反向代换、A的反矩阵、矩阵分解A=LU)
- 方阵Ax = b 之全解(包括含 b 之行向量空间、A之秩、A的核空间及列简化后之 Ax = 0 的特别解)。
- 基底与维度。(四种基本子空间的基础)
- 最小平方法(由投影观念求最近线)
- Gram-Schmidt正交化(A = QR的分解法)
- 行列式的性质(导向余因子方程及n! 种排列之和、inv(A)的应用及求体积)
- 固有值与固有向量(A的对角化、计算A^k的幂及矩阵指数以解差分与微分方程)
- 对称矩阵与正交矩阵(实数固有值、正交固有向量与x'Ax > 0 检定等及其应用)
- 线性转换与基底变换(与奇异值分解法连结 - 以正交化为基准来对角化A)
- 线性代数在工程学上的应用(图形与网络、Markov矩阵、Fourier、快速Fourier转换、线性规划)
作业演练
作业演练为修习线性代数的必要历程。这些作业并非考试；我们鼓励学生们向难题挑战，而“难题”的定义因人而异。讨论是学习线性代数的一种健康方法。请各位以自己的对问题之理解与方法完成演练。
学科测验
本门课会有三次为时一小时的期中测验。测验时不许使用计算机及参考笔计。
成绩评量
作业演练 24%
三次期中考 42%
期末考 34%
MATLAB®
若干作业会要求以 MATLAB®完成。 MATLABR®是线性代数的优异工具，本课程将以此工具为大部份的作业命题。MATLABR 的学生版己升级至 MATLABR version 5 ，其中包括了极佳的绘图功能。
课程录影
本网站亦提供Strang教授溯自1999年的授课录影(详参课程网页)
Text
Introduction to Linear Algebra 3rd Edition by Gilbert Strang, Wellesley-Cambridge Press (March 2003).
Goals of the Linear Algebra Course
The goals for 18.06 are *using matrices and also understanding them*
Here are key computations and some of the ideas behind them:
- Solving Ax = b for square systems by elimination (pivots, multipliers,
back substitution, invertibility of A, factorization into A = LU)
- Complete solution to Ax = b (column space containing b, rank of A,
nullspace of A and special solutions to Ax = 0 from row reduced R)
- Basis and dimension (bases for the four fundamental subspaces)
- Least squares solutions (closest line by understanding projections)
- Orthogonalization by Gram-Schmidt (factorization into A = QR)
- Properties of determinants (leading to the cofactor formula and
the sum over all n! permutations, applications to inv(A) and volume)
- Eigenvalues and eigenvectors (diagonalizing A, computing powers A^k
and matrix exponentials to solve difference and differential equations)
- Symmetric matrices and positive definite matrices (real eigenvalues
and orthogonal eigenvectors, tests for x'Ax > 0, applications)
- Linear transformations and change of basis (connected to the Singular
Value Decomposition -- orthonormal bases that diagonalize A)
- Linear algebra in engineering (graphs and networks, Markov matrices,
Fourier matrix, Fast Fourier Transform, linear programming)
Homework
The homeworks are essential in learning linear algebra. They are not a test and you are encouraged to talk to other students about difficult problems-after you have found them difficult. Talking about linear algebra is healthy. But you must write your own solutions.
Exams
There will be three one-hour exams at class times and a final exam. The use of calculators or notes is not permitted during the exams.
Your Grade
Problems sets 24%
Three one-hour exams 42%
Final exam 34%
MATLAB®
Some homework problems will require you to use MATLAB®. MATLAB® is the outstanding software for linear algebra. 18.06 will use it for the best homework problems. The student version of MATLAB® is now upgraded to MATLAB® version 5 with great graphics.
Videos
Videos of Professor Strang's lectures from 1999 are available on the web (see the course web page).
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