主题目录

  • 课程介绍( Course Introduction)

    • University:   Massachusetts Institute of Technology

      Instructors:  Dr. Dan Ciubotaru

      Course Number:  18.700

      Level:  Undergraduate

      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.

  • 教学大纲(Syllabus)

    • Prerequisites

      Multivariable Calculus (18.02)

  • 其他教学资源(Other Resources)

    • Readings

      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.

      Lec #TOPICSReadings
      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