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)
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.
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.
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 of Professor Strang's lectures from 1999 are available on the web (see the course web page).