MATH 2568: Linear Algebra

Matrix algebra, vector spaces and linear maps, bases and dimension, eigenvalues and eigenvectors, applications.

Required Textbook: Introduction to Linear Algebra by Lee W. Johnson, R. Dean Riess and Jimmy T. Arnold, 5th edition, 2017, Pearson

Learning Outcomes

By the end of this course, successful linear algebra students should be able to:

• Understand algebraic and geometric representations of vectors in $\mathbb{R}^n$ and their operations, including addition, scalar multiplication and dot product. understand how to determine the angle between vectors and the orthogonality of vectors.
• Solve systems of linear equations using Gauss-Jordan elimination to reduce to echelon form. Solve systems of linear equations using the inverse of the coefficient matrix when possible. Interpret existence and uniqueness of solutions geometrically.
• Perform common matrix operations such as addition, scalar multiplication, multiplication, and transposition. Discuss associativity and noncommutativity of matrix multiplication.
• Discuss spanning sets and linear independence for vectors in $\mathbb{R}^n$. For a subspace of $\mathbb{R}^n$, prove all bases have the same number of elements and define the dimension. Prove elementary theorems concerning rank of a matrix and the relationship between rank and nullity.
• Interpret a matrix as a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$. Discuss the transformation’s kernel and image in terms of nullity and rank of the matrix. Understand the relationship between a linear transformation and its matrix representation, and explore some geometric transformations in the plane. Interpret a matrix product as a composition of linear transformations.
• Use determinants and their interpretation as volumes. Describe how row operations affect the determinant. Analyze the determinant of a product algebraically and geometrically.
• Define eigenvalues and eigenvectors geometrically. Use characteristic polynomials to compute eigenvalues and eigenvectors. Use eigenspaces of matrices, when possible, to diagonalize a matrix.
• Use axioms for abstract vector spaces (over the real or complex fields) to discuss examples (and non-examples) of abstract vector spaces such as subspaces of the space of all polynomials.