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	Introduction to Linear Systems of Equations	Adding equations and multiplying equations by constants Solving 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	Introduction to Linear Systems of Equations Equation for a line	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	1.8 and 1.9	2.1	Introduction to Linear Transformations	Introduction to Linear Systems of Equations Vectors, Matrices, and Gauss-Jordan Elimination Transformations of Functions	Linear Transformation Requirements for a transformation to be linear
Week 4	2.1	2.3	Matrix Algebra and Matrix Multiplication	Range of a Function Transformations of Functions	Matrix Operations Matrix products by columns Matrix products using the dot product
Week 4	2.2 and 2.3	2.4	The Inverse of a Linear Transformation	Matrix Algebra and Matrix Multiplication Inverse Functions Introduction to Linear Systems of Equations	The Identity matrix The Inverse of a Matrix The Inverse of a Linear Transformation Various characterizations for an invertible matrix
Week 5	1.7, 2.8, and 2.9	3.2	Subspaces of Rⁿ and Linear Independence	Matrix Algebra and Matrix Multiplication The Inverse of a Linear Transformation	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	Matrix Algebra and Matrix Multiplication Subspaces of Rⁿ and Linear Independence	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	Introduction to Linear Transformations Range of a Function The Inverse 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	Introduction to Vector Spaces Subspaces of Rⁿ and Linear Independence	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 6.4	Appendix A and 5.1	Dot Products and Orthogonality	The Dimension of a Vector Space Subspaces of Rⁿ and Linear Independence	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	Subspaces of Rⁿ and Linear Independence Dot Products and Orthogonality	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	Dot Products and Orthogonality The Column Space and Nullspace of a Linear Transformation	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	Orthonormal Bases and the Gram-Schmidt Process The Inverse of a Linear Transformation Signum Notation	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	Orthonormal Bases and the Gram-Schmidt Process Introduction to Determinants The Inverse of a Linear Transformation	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	Introduction to Determinants The Column Space and Nullspace of a Linear Transformation The Inverse of a Linear Transformation	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	Eigenvalues and Eigenvectors The Column Space and Nullspace of a Linear Transformation	Similar matrices Diagonalization in terms of linearly independent eigenvectors Algebraic and geometric multiplicity for a specific eigenvalue The strategy for diagonalization

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	1.1, 1.2	Systems of Linear Equations	Adding and multiplying equations by constants Solving Equations	Vectors and Matrices Gauss-Jordan elimination
Week 2	1.3	Solutions of Linear Systems	Gauss-Jordan elimination Equation for a line	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	Range of a Function Transformations of Functions	Linear transformations and their properties Geometry of Linear Transformations (rotations, scalings and projections)
Week 4	2.3 and 2.4	Matrix Products and Inverses	Linear Combinations Inverse Functions Vectors and the Inner product	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	Solutions of Linear Systems Image of a Function Kernel of a Function	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	Linear Independence	Solutions of Linear Systems Image and Kernel of a Linear Transform	Subspaces of Rⁿ Redundant vectors and linear independence Characterizations of Linear Independence
Week 6	3.2	Bases of Subspaces	Linear Independence The span of a set of vectors	Bases and Linear independence Basis of the image Basis and unique representation
Week 5	3.3	The Dimension of a Subspace	Image of a Function Bases and Linear Independence Linear transformations	Dimension of the Image Rank-nullity theorem Various bases in Rn
Week 7/8	3.4	Similar Matrices and Coordinates	Bases of Subspaces Equivalence Relations	Coordinates in a subspace of Rⁿ Similar matrices Diagonal matrices
Week 9	5.1	Orthogonal Projections and Orthonormal Bases	Parallel and Perpendicular Lines Absolute value function Trig. Functions: Unit Circle Approach Inner Products Bases and Linear Independence	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	Unit vectors Inner Products Orthonormal Bases Bases and Linear Independence	Gram-Schmidt process QR Factorization
Week 11	5.3	Orthogonal Transformations and Orthogonal Matrices	Image and Kernel of a Linear Transform Inner Products Orthogonal Projections and Orthonormal Bases	Orthogonal Transformations Properties of Othogonal Transformations Transpose of a Matrix The matrix of an Orthogonal Projection
Week 11	5.3	Least Squares	Linear transformations Orthogonal Transformations and Orthogonal Matrices Orthogonal Projections	The Least Squares Solution The Normal Equation Another matrix for an Orthogonal Projection
Week 11	6.1 and 6.2	Determinants	Summation Notation Sgn Function Inverse of a Linear Transformation Transpose of a Matrix	Properties of Determinants Sarrus's Rule Row operations and determinants Invertibility based on the determinant
Week 12	6.3	Cramer's Rule	Determinants Invertible matrices Rotations	Parrallelepipeds in Rn Geometric Interpretation of the Determinant Cramer's rule
Week 13	7.1	Diagonalization	Similar Matrices and Coordinates Orthogonal Transformations and Orthogonal Matrices	Diagonalizable matrices Eigenvalues and eigenvectors Real eigenvalues of orthogonal matrices
Week 14	7.2 and 7.3	Finding Eigenvalues and Eigenvectors	Determinants Invertible matrices Diagonalization Image and Kernel of a Linear Transform	Eigenvalues from the characteristic equation Eigenvalues of Triangular matrices Characteristic Polynomial Eigenspaces and eigenvectors Geometric and algebraic multiplicity Eigenvalues of similar matrices
Week 14	8.1	Symmetric Matrices	Similar Matrices and Coordinates Transpose of a Matrix Eigenvalues and Eigenvectors Algebraic and Geometric Multiplicities	Orthogonally Diagonalizable Matrices Spectral Theorem The real eigenvalues of a symmetric matrix
Week 14	8.2	Quadratic Forms	Symmetric Matrices Finding Eigenvalues and Eigenvectors Conics	Quadratic Forms Diagonalizing a Quadratic Form Definiteness of a Quadratic Form Principal Axes Ellipses and Hyperbolas from Quadratic Forms