This is an archived course. A more recent version may be available at ocw.mit.edu.

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Readings

All of the reading assignments below refer to the main textbook:

Buy at Amazon Jacob, Bill. Linear Algebra. New York, NY: W.H. Freeman, 1990. ISBN: 0716720310. (Out of print.)

There are two other recommended books for this course:

Buy at Amazon Hoffman, K., and R. Kunze. Linear Algebra. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1971. ISBN: 0135367972.

Buy at Amazon Bretscher, O. Linear Algebra with Applications. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2004. ISBN: 0131453343.

Lec # TOPICS Readings
1 Systems of Linear Equations Section 0.1: Systems of linear equations, row equivalence
2 Echelon Form Section 0.2: Gaussian and Gauss-Jordan elimination, (reduced) row-echelon form, back-substitution
3 Matrices Section 0.3: Matrices, matrix operations, block multiplication
4 Matrices (cont.) Section 0.4: Matrices and linear systems, elementary matrices, (reduced) row-echelon matrices
5 Solution Spaces Section 0.5: The space of solutions to a homogeneous linear system, uniqueness of the reduced row-echelon form, matrix rank, criterion for existence of solutions
6 Inverses and Transposes Section 0.6: Matrix inverses (right, left), invertible matrices, transpose of a matrix, symmetric matrices
7 Fields and Spans Section 1.1: Definition of a field F, examples: Q, R, C, Z/pZ (see also 1.6. pp. 132-133), linear combinations of vectors, and spans in Fn
8 Vector Spaces Section 1.2: Vector spaces, definition and examples, sub-spaces, the row space, column space, and nullspace of a matrix
9 Linear Independence Section 1.3: Linear independent vectors
10 Basis and Dimension Section 1.4: Basis of a vector space, dimension, bases for the row space and column space of a matrix, Rank plus nullity theorem for matrices, Basis extension theorem
11 Coordinates Section 1.5: Coordinates with respect to an ordered basis, change of coordinates matrix
12 Review for Quiz 1  
13 Quiz 1 (Chapters 0-1)  
14 Determinants Section 2.1 (pp. 137-143): Determinant function (definition, properties, uniqueness), computing determinants using row-reduction

Section 2.1 (pp. 144-146): invertible matrices, det(AB) = det(A)det(B), det(At) = det (A)
15 Permutations Section 2.2: Permutations and the permutation definition of the determinant
16 Determinants (cont.) Section 2.2: Permutations and the permutation definition of the determinant
17 Laplace Expansion Section 2.3: Cofactor (Laplace) expansion of the determinant, the adjoint of a matrix, finding the inverse using the adjoint, Cramer's rule
18 Review for Quiz 2  
19 Quiz 2 (Chapter 2)  
20 Linear Transformations Section 3.1: Linear transformations (definition, examples), matrix associated to a linear transformation
21 Rank, Kernel, Image Section 3.2: Properties of linear transformations, rank, kernel, image, Rank plus nullity theorem for linear transformations, one-one, onto, isomorphism
22 Matrix Representations Section 3.3: Matrix representations for linear transformations, similar matrices
23 Eigenspaces Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial
24 Eigenspaces (cont.) Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial
25 Diagonalization Section 3.5: Diagonalizable linear operators and matrices
26 Cayley-Hamilton Theorem Section 6.1: Cayley-Hamilton theorem, minimal polynomial
27 Jordan Canonical Form Section 6.4 (pp. 373-376): Jordan form, generalized eigenvectors and Primary decomposition theorem from section 6.5 (see also J. Starr's notes from Fall 2004)
28 Review for Quiz 3  
29 Quiz 3 (Chapter 3)  
30 Computing Generalized Eigenvectors Section 6.4 (pp. 376-384): More on Jordan form, computing generalized eigenvectors
31 Norms and Inner Products Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality
32 Norms and Inner Products (cont.) Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality
33 Orthogonal Bases Section 4.2: Orthogonal and orthonormal bases, Gram-Schmidt algorithm, QR decomposition
34 Orthogonal Projections Section 4.3: Orthogonal projections, orthogonal complement, direct sums
35 Isometries, Spectral Theory Section 4.5 (pp. 282-285): Isometries, orthogonal and unitary matrices
36 Singular Value Decomposition Section 4.5 (pp. 286-291): Self-adjoint operators, symmetric and hermitian matrices, eigenvalues of self-adjoint operators, Principal axis theorem, Spectral resolution
37 Polar Decomposition Section 4.6: Singular value decomposition, positive (semi)definite matrices, Polar decomposition
38 Review for the Final