Linear Algebra IV: Orthogonality & Symmetric Matrices and the SVD

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
  points
• 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