## 主题目录

### Course Description

This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with Linear Algebra (18.06), more emphasis is placed on theory and proofs.

• ## Prerequisites

Multivariable Calculus (18.02)

• ### 其他教学资源(Other Resources)

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

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:

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

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

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
• Study materials