Linear Algebra III: Determinants and Eigenvalues

Upon completion of this course, learners will be able to:

• Compute determinants of using cofactor expansions and properties of determinants
• Compute the area of regions in R^3 under a given linear transformation using determinants
• Model and solve real-world problems using Markov chains
• Verify that a given vector is an eigenvector of a matrix
• Verify that a scalar is an eigenvalue of a matrix
• Construct an eigenspace for a matrix
• Characterize the invertibility of a matrix using determinants and eigenvalues
• Apply theorems related to eigenvalues (for example, to characterize the invertibility of a matrix)
• Factorize 2 × 2 matrices that have complex eigenvalues
• Use eigenvalues to determine identify the rotation and dilation of a linear transform
• Apply theorems to characterize matrices with complex eigenvalues
• Apply matrix powers and theorems to characterize the long-term behavior of a Markov chain
• Construct a transition matrix, a Markov Chain, and a Google Matrix for a given web, and compute the PageRank of the web.