# MAT2233

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
• 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
• 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 Operations
• Matrix products by columns
• Matrix products using the dot product
Week 3
2.2 and 2.3
2.4
• The Identity matrix
• The Inverse of a Matrix
• Various characterizations for an invertible matrix

Week 4
1.8 and 1.9
2.1
• Linear Transformations
• Requirements for a transformation to be linear
Week 5
1.7, 2.8, and 2.9
3.2
• 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
• 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 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 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
• 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
• 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 orthogonal complement of the image is equal to the left nullspace (or kernel 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
• 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

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