MAT2233

From Department of Mathematics at UTSA
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A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of Linear Algebra and its history.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
1.1, 1.2

Systems of Linear Equations

  • Vectors and Matrices
  • Gauss-Jordan elimination
Week 2
1.3

Solutions of Linear Systems

  • Rank of a matrix
  • Matrix addition
  • The product Ax (where A is a matrix and x is a vector)
  • The Inner product
  • Linear Combinations


Week 3
2.1 and 2.2

Linear Transformations

  • Linear transformations and their properties
  • Geometry of Linear Transformations (rotations, scalings and projections)
Week 4
2.3 and 2.4

Matrix Products and Inverses

  • Matrix Products (both inner product and row-by-column methods)
  • The Inverses of a linear transform


Week 6
3.1

Image and Kernel of a Linear Transform

  • The image of a Linear transformation
  • The kernel of a linear transformation
  • Span of a set of vectors
  • Alternative characterizations of Invertible matrices


Week 6
3.2

Bases and Linear Independence

  • Subspaces in Different Dimensions
  • Bases and Linear independence
  • Characterizations of Linear Independence


Week 5
3.3

The Dimension of a Subspace

  • Dimension of the Image
  • Rank-nullity theorem
  • Various bases in Rn


Week 7/8
3.4


Similar Matrices and Coordinates

  • Coordinates in a subspace of Rn
  • Similar matrices
  • Diagonal matrices


Week 9
5.1

Orthogonal Projections and Orthonormal Bases

  • Magnitude (or norm or length) of a vector
  • Unit Vectors
  • Cauchy-Schwarz Inequality
  • Orthonormal vectors
  • Orthogonal complement
  • Orthogonal Projection
  • Orthonormal bases
  • Angle between vectors


Week 10
5.2


Gram-Schmidt Process and QR Factorization

  • Gram-Schmidt process
  • QR Factorization


Week 11
5.3

Orthogonal Transformations and Orthogonal Matrices

  • Orthogonal Transformations
  • Properties of Othogonal Transformations
  • Transpose of a Matrix
  • The matrix of an Orthogonal Projection


Week 11
5.3

Least Squares

  • The Least Squares Solution
  • The Normal Equation
  • Another matrix for an Orthogonal Projection


Week 11
6.1 and 6.2

Determinants

  • Properties of Determinants
  • Sarrus's Rule
  • Row operations and determinants
  • Invertibility based on the determinant


Week 12
6.3


Cramer's Rule

  • Parrallelepipeds in Rn
  • Geometric Interpretation of the Determinant
  • Cramer's rule


Week 13
7.1


Diagonalization

  • Diagonalizable matrices
  • Eigenvalues and eigenvectors
  • Real eigenvalues of orthogonal matrices


Week 14
7.2 and 7.3

Finding Eigenvalues and Eigenvectors

  • Eigenvalues from the characteristic equation
  • Eigenvalues of Triangular matrices
  • Characteristic Polynomial
  • Eigenspaces and eigenvectors
  • Geometric and algebraic multiplicity
  • Eigenvalues of similar matrices


Week 14
7.2 and 7.3

Symmetric Matrices

  • Orthogonally Diagonalizable Matrices
  • Spectral Theorem
  • The real eigenvalues of a symmetric matrix
Week 14
8.2

Quadratic Forms

  • Quadratic Forms
  • Diagonalizing a Quadratic Form
  • Definiteness of a Quadratic Form
  • Principal Axes
  • Ellipses and Hyperbolas from Quadratic Forms