Prerequisite

Calculus of several variables (18.02)

Text

Strang, Gilbert. Introduction to Linear Algebra. 3rd ed. Wellesley, MA: Wellesley-Cambridge Press, March 2003. ISBN: 0961408898.

Goals

The goals for 18.06 are using matrices and also understanding them.

Here are key computations and some of the ideas behind them:

1. Solving Ax = b for square systems by elimination (pivots, multipliers, back substitution, invertibility of A, factorization into A = LU)
2. 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)
3. Basis and dimension (bases for the four fundamental subspaces)
4. Least squares solutions (closest line by understanding projections)
5. Orthogonalization by Gram-Schmidt (factorization into A = QR)
6. Properties of determinants (leading to the cofactor formula and the sum over all n! permutations, applications to inv(A) and volume)
7. Eigenvalues and eigenvectors (diagonalizing A, computing powers A^k and matrix exponentials to solve difference and differential equations)
8. Symmetric matrices and positive definite matrices (real eigenvalues and orthogonal eigenvectors, tests for x'Ax > 0, applications)
9. Linear transformations and change of basis (connected to the Singular Value Decomposition -- orthonormal bases that diagonalize A)
10. 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.

Grading

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.