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指定教材
《線性代數導論》(Introduction to Linear Algebra)第三版(2003年三月)，Gilbert Strang著，Wellesley-Cambridge Press出版。
線性代數課程目標 e
本課程(18.06)的目的為使學生了解矩陣的性質並應用。
下列為其重要演算及其所本之觀念：
- 以消去法解方陣 Ax = b 。(包括高斯消去法、乘數消去法、反向代換、A的反矩陣、矩陣分解A=LU)
- 方陣Ax = b 之全解(包括含 b 之行向量空間、A之秩、A的核空間及列簡化後之 Ax = 0 的特別解)。
- 基底與維度。(四種基本子空間的基礎)
- 最小平方法(由投影觀念求最近線)
- Gram-Schmidt正交化(A = QR的分解法)
- 行列式的性質(導向餘因子方程及n! 種排列之和、inv(A)的應用及求體積)
- 固有值與固有向量(A的對角化、計算A^k的冪及矩陣指數以解差分與微分方程)
- 對稱矩陣與正交矩陣(實數固有值、正交固有向量與x'Ax > 0 檢定等及其應用)
- 線性轉換與基底變換(與奇異值分解法連結 - 以正交化為基準來對角化A)
- 線性代數在工程學上的應用(圖形與網絡、Markov矩陣、Fourier、快速Fourier轉換、線性規劃)
作業演練
作業演練為修習線性代數的必要歷程。這些作業並非考試；我們鼓勵學生們向難題挑戰，而「難題」的定義因人而異。討論是學習線性代數的一種健康方法。請各位以自己的對問題之理解與方法完成演練。
學科測驗
本門課會有三次為時一小時的期中測驗。測驗時不許使用計算機及參考筆計。
成績評量
作業演練 24%
三次期中考 42%
期末考 34%
MATLAB®
若干作業會要求以 MATLAB®完成。 MATLABR®是線性代數的優異工具，本課程將以此工具為大部份的作業命題。MATLABR 的學生版己升級至 MATLABR version 5 ，其中包括了極佳的繪圖功能。
課程錄影
本網站亦提供Strang教授溯自1999年的授課錄影(詳參課程網頁)
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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