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Linear Algebra IV: Orthogonality & Symmetric Matrices and the SVD
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In the first part of this course you will explore methods to compute an approximate solution to an inconsistent system of equations that have no solutions. Our overall approach is to center our algorithms on the concept of distance. To this end, you will first tackle the ideas of distance and orthogonality in a vector space. You will then apply orthogonality to identify the point within a subspace that is nearest to a point outside of it. This has a central role in the understanding of solutions to inconsistent systems. By taking the subspace to be the column space of a matrix, you will develop a method for producing approximate (“least-squares”) solutions for inconsistent systems.
You will then explore another application of orthogonal projections: creating a matrix factorization widely used in practical applications of linear algebra. The remaining sections examine some of the many least-squares problems that arise in applications, including the least squares procedure with more general polynomials and functions.
This course then turns to symmetric matrices. arise more often in applications, in one way or another, than any other major class of matrices. You will construct the diagonalization of a symmetric matrix, which gives a basis for the remainder of the course.
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Upon completion of this course, learners will be able to:
- Compute dot product of two vectors, length of a vector, distance between points, and angles between vectors
- Apply theorems related to orthogonal complements, and their relationships to Row and Null
space, to characterize vectors and linear systems
- Compute orthogonal projections and distances to express a vector as a linear combination of orthogonal vectors, construct vector approximations using projections, and characterize bases for subspaces, and construct orthonormal bases
- Apply the iterative Gram Schmidt Process, and the QR decomposition, to construct an orthogonal basis
- Construct the QR factorization of a matrix
- Characterize properties of a matrix using its QR decomposition
- Compute general solutions and least squares errors to least squares problems using the normal
equations and the QR decomposition
- Apply least-squares and multiple regression to construct a linear model from a set of data
- Apply least-squares to fit polynomials and other curves to data
- Construct an orthogonal diagonalization of a symmetric matrix
- Construct a spectral decomposition of a matrix