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Linear Algebra III: Determinants and Eigenvalues
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At the beginning of this course we introduce the determinant, which yields two important concepts that you will use in this course. First, you will be able to apply an invertibility criterion for a square matrix that plays a pivotal role in, for example, the understanding of eigenvalues. You will also use the determinant to measure the amount by which a linear transformation changes the area of a region. This idea plays a critical role in computer graphics and in other more advanced courses, such as multivariable calculus.
This course then moves on to eigenvalues and eigenvectors. The goal of this part of the course is to decompose the action of a linear transformation that may be visualized. The main applications described here are to discrete dynamical systems, including Markov chains. However, the basic concepts— eigenvectors and eigenvalues—are useful throughout industry, science, engineering and mathematics.
Prospective students enrolling in this class are encouraged to first complete the linear equations and matrix algebra courses before starting this class.
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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.