Catalog description: A study of vectors, matrices and linear transformations, principally in two and three dimensions, including treatments of systems of linear equations, determinants, and eigenvalues. Prerequisite: MAT 1120 or permission of the instructor.
I appreciate linear algebra for its versatility. I have friends from graduate school who design commercial airplanes, friends who use math to protect our environment, friends who fight disease with mathematical models, and friends who work to protect our national security. They all use linear algebra regularly!
My own research centers on topics from linear algebra. I’ve worked with students to study voting theory, music theory, and scheduling theory. We've also studied more abstract topics including symmetric spaces as well as Lie and Leibniz algebras. Whether we are thinking in terms of applications or more concerned with theory, our need for the tools of linear algebra remains constant.
There’s so much to learn that we can’t cover all of it in one semester. But, I’m extremely excited to be able to give you the best introduction to linear algebra that I can. The following learning outcomes detail the skills you will build throughout our course.
Students will ...
link fundamental structures of linear algebra including systems of linear equations, vectors, matrices, spaces, and linear transformations
connect fundamental linear algebra concepts including determinants and eigenvalues to multiple structures listed above
implement algebraic skills relevant to linear algebra
demonstrate spatial visualization skills in two and three dimensions
employ appropriate software to solve linear algebra problems
investigate applications of linear algebra
communicate linear algebra in written and/or oral forms
Students will...
link algebra and geometry of systems of equations and their solutions (CLO 3,4)
apply elementary row operations to transform matrices into row echelon forms (CLO 3)
determine solutions to systems of linear equations by hand and in Maple (CLO 3,4,7)
link matrix equations, vector equations, and systems of equations (CLO 1,3,4)
classify linear systems as homogeneous or non-homogeneous as well as consistent or inconsistent (CLO 3,4)
express infinite solution sets in parametric form (CLO 1,3,4)
compute sums, scalar products, and linear combinations of vectors, and the product of a matrix and a vector (CLO 3,4)
characterize linear combinations of vectors and the span of a set of vectors, both algebraically and geometrically (CLO 1,3,4)
determine if a set of vectors is linearly independent (CLO 1,3,4)
explore applications of systems of linear equations (CLO 6,7)
Students will ...
determine scalar multiples, sums and products of matrices as well as the transpose of a matrix (CLO 3, 4, 5)
determine if a given matrix is invertible and if so find the inverse of that matrix (CLO 3,4,5)
connect matrix multiplication to matrix-vector products and row operations (CLO 1, 3, 5)
apply the inverse matrix theorem (CLO 1, 3, 4, 5)
apply algorithms including cofactor expansion to find determinants (CLO 2, 3, 5)
investigate geometric interpretations of determinants (CLO 1, 2, 3, 4, 5)
characterize transformations as linear or nonlinear using the definition (CLO 1, 3, 4)
describe the standard matrix of a linear transformation (CLO 1, 3, 4)
interpret linear transformations in R^2 and R^3 both algebraically as matrix-vector products and geometrically in terms of actions on the plane and 3-space. (CLO 1,3,4,5)
connect function composition and inverse functions for linear transformations to matrix multiplication and inverse matrices (CLO 1, 3, 4, 5)
connect kernel and range of linear transformations to uniqueness and existence of solutions to linear systems (CLO 1, 3, 4)
explore applications of matrix algebra and linear transformations including applications to computer graphics (CLO 6,7)
Students will ...
Compute norms and inner products in R^n (CLO 3, 4, 5)
Determine if vectors are orthogonal in R^n (CLO 3, 4, 5)
Determine whether a given set is a subspace (CLO 1, 3, 4, 5)
Apply definitions of null and column spaces of a matrix (CLO 1, 2, 6)
Determine basis and dimension for subspaces of R^n, particularly the column space and null space of a matrix (CLO 3, 4, 5)
Connect subspaces to the rank-nullity theorem and the invertible matrix theorem (CLO 1, 2)
Determine eigenvalues, eigenvectors, and the characteristic polynomial by hand and in Maple. (CLO 2, 3, 4, 5, 7)
Determine bases for eigenspaces. (CLO 1, 2, 3, 4)
Link algebra and geometry of eigenvalues and eigenvectors (CLO 2, 3)
Characterize the long-term behavior of dynamical systems using eigenvalue decompositions (CLO 1, 2, 3, 4, 6).
explore applications of vector subspaces, orthogonality, and eigenspaces (CLO 6,7)