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Pathway 05 · University

Linear Algebra and Matrices

Vector spaces, linear maps, matrices and eigen-structure, including spaces of functions and the matrix of differentiation. The path ends with the Matrix Boundary Method described in chapter 5 of the book.

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Audience
University students in linear algebra courses, and students of analysis or differential equations who want to see calculus operations written as matrices.
Modules
6, in order
Level
University

Prerequisites

  • Solving systems of linear equations
  • University Foundations-level algebra
  • Basic differentiation, for the function-space modules

You will be able to

  • Decide linear independence, and find a basis and the dimension of a subspace, including subspaces of functions
  • Row-reduce a matrix, compute its rank and null space, and describe all solutions of Ax=bAx = b
  • Compute determinants and inverses and relate invertibility to rank and to eigenvalues
  • Find eigenvalues and eigenvectors, diagonalize a matrix when possible, and recognize a defective matrix
  • Write the matrix of differentiation on a space such as span⁡{ex,xex}\operatorname{span}\{e^{x}, xe^{x}\} and use its inverse to find an antiderivative
  • Compute etAe^{tA} for a 2×22\times 2 matrix and use it to solve x′=Axx' = Ax

Modules

Work through these in order.

  1. 01

    Vectors, systems and elimination

    Solve linear systems systematically and read the structure of their solution sets.

    • Gaussian elimination
    • Row echelon form
    • Rank
    • Homogeneous and nonhomogeneous systems
  2. 02

    Vector spaces, bases and dimension

    Abstract the structure of systems into vector spaces, including spaces of functions.

    • Subspaces and span
    • Linear independence
    • Basis and dimension
    • Function spaces
  3. 03

    Linear maps and their matrices

    Represent linear maps by matrices relative to chosen bases.

    • Kernel and image
    • Matrix of a linear map
    • Change of basis
    • Differentiation as a linear map
  4. 04

    Determinants and eigen-structure

    Use determinants and eigenvalues to understand what a matrix does.

    • Determinants
    • Eigenvalues and eigenvectors
    • Diagonalization
    • Jordan blocks in small cases
  5. 05

    Matrix exponential and linear systems

    Connect linear algebra to differential equations through the matrix exponential.

    • Definition of eAe^{A}
    • Computing via diagonalization
    • Solving x′=Axx' = Ax
    • Stability from eigenvalues
  6. 06

    Differentiation matrices and the Matrix Boundary Method

    Read chapter 5 of the book: finite-dimensional spaces closed under differentiation, the matrix of differentiation, and the book's Matrix Boundary Method for truncated Laplace-type integrals and linear ODEs.

    • Closure under differentiation
    • Coordinate vectors of functions
    • Representing (D+b)−1(D+b)^{-1} by a matrix
    • MBM worked examples

Next step

Study this pathway with guidance.

Rida can follow this pathway with you one-to-one or in a small group, with diagnostics and feedback.

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Supporting material

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Read in the Journal

Research context

In the book

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