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 from Lay Sections from Bretscher Topics Prerequisite Skills Student Learning Outcomes
Week 1
1.1 and 1.2
1.1

Introduction to Linear Systems of Equations

  • Using elimination to find solutions of linear systems
  • The Geometrical interpretation of solutions to linear systems
Week 2
1.3, 1.4, and 1.5
1.2 and 1.3

Vectors, Matrices, and Gauss-Jordan Elimination

  • Vectors and vector spaces
  • Matrix notation
  • The Gauss-Jordan method for solving a linear system of equation
  • The rank of a matrix
  • Sums of Matrices
  • The product Ax (where A is a matrix and x is a vector)
  • The Dot product
  • Linear Combinations


Week 3
2.1
2.3

Matrix Algebra and Matrix Multiplication

  • Matrix Operations
  • Matrix products by columns
  • Matrix products using the dot product
Week 3
2.2 and 2.3
2.4

The Inverse of a Linear Transformation

  • The Identity matrix
  • The Inverse of a Matrix
  • The Inverse of a Linear Transformation
  • Various characterizations for an invertible matrix


Week 4
1.8 and 1.9
2.1

Introduction to Linear Transformations

  • Linear Transformation
  • Requirements for a transformation to be linear


Week 5
1.7, 2.8, and 2.9
3.2

Subspaces of Rⁿ and Linear Independence

  • Definition of a subspace of Rⁿ
  • Defining linear independence for a set of vectors
  • Definition of a basis for a subspace


Week 6
4.1
4.1

Introduction to Vector Spaces

  • Definition of a vector space(or linear space)
  • Subspaces of vector spaces
  • Linear combinations and bases for vector spaces
  • Examples of vector spaces of functions


Week 6
4.2
3.1

The Column Space and Nullspace of a Linear Transformation

  • The image (or column space) of a linear transformation
  • The kernel (or nullspace) of a linear transformation
  • Properties of the kernel


Week 7
4.3 and 4.5
3.3 and 4.1

The Dimension of a Vector Space


  • The number of vectors in a basis of Rn
  • Dimension of a subspace in Rⁿ
  • The dimension of a vector space
  • The dimension of the nullspace (or kernel) and the column space (or image)
  • The Rank-nullity Theorem


Week 8
6.1 and 6.2
Appendix A and 5.1

Dot Products and Orthogonality

  • Orthogonal vectors
  • Length (or magnitude or norm) of a vector
  • Unit vectors
  • Orthonormal vectors
  • Orthogonal projections
  • Orthogonal complements
  • Cauchy-Schwarz inequality
  • The angle between vectors
Week 9
6.3 and 6.4
5.2 and 5.3

Orthonormal Bases and the Gram-Schmidt Process

  • Orthogonal transformations
  • Orthonormal Bases
  • Orthogonal matrices
  • The transpose of a matrix
  • The Gram-Schmidt Process
  • QR factorization
Week 10
6.5 and 6.6
5.4

The Least-squares Solution

  • The orthogonal complement of the image is equal to the left nullspace (or kernal of the transpose) for all matrices
  • The least-squares solution for a linear system
  • Data fitting using the least-squares solution


Week 11
3.1 and 3.2
6.1 and 6.2

Introduction to Determinants

  • The determinant of 2 by 2 and 3 by 3 matrices
  • The determinant of a general n by n matrix
  • The determinant of a triangular matrix
  • Properties of the determinant
  • The determinant of the transpose
  • Invertibility and the determinant


Week 12
3.3
6.3

The Geometric Interpretation of the Determinant


  • Cramer's Rule
  • The adjoint and inverse of a matrix
  • The area of a parallelogram and the volume of a parallelepiped


Week 13
The beginning of 5.3 as well as the sections 5.1 and 5.2
7.1, 7.2 and the beginning of 7.3

Eigenvalues and Eigenvectors

  • The requirement for a matrix to be diagonalizable
  • Definition of an eigenvector
  • The characteristic equation used to find eigenvalues
  • Eigenvalues of a triangular matrix
  • Eigenspaces for specific eigenvalues


Week 14
5.3 and 5.4
3.4 and 7.3

Diagonalization of Matrices

  • Similar matrices
  • Diagonalization in terms of linearly independent eigenvectors
  • Algebraic and geometric multiplicity for a specific eigenvalue
  • The strategy for diagonalization