# Matrices & Geometry

Introductory course emphasizing techniques of linear algebra with application to engineering; topics include matrix operations, determinants, linear equations, vector spaces, linear transformations, eigenvalues, eigenvectors, inner products and norms, orthogonality, equilibrium, and linear dynamical systems.

You may purchase a copy of our courseware without any instructional services such as mentor feedback or grading if you are simply seeking reference materials rather than enrolling for a class.

**Syllabus**

**Lesson 1: PlotFest**

Using Mathematica to plot in two and three dimensions with special attention to parametric plotting.

**Lesson 2: Perpendicular Frames in 2D and 3D**

• Vectors in 2D and vectors in 3D

• Addition and subtraction of vectors

• Dot product and Cross product

• Aligning and hanging on perpendicular frames to plot tilted ellipses and ellipsoids

• Right hand frames versus left hand frames

• Resolution of vectors into perpendicular components

• Planes and lines through the origin

**Lesson 3: 2D Matrix Action**

• Matrix multiplication

• Hitting the unit circle with a matrix and observing the result through matrix action movies

• Linearity of matrix multiplication

• Taking a 2D perpendicular frame and using it to plot tilted ellipses

• Rotation matrices and right hand frames

• Reflection matrices and left hand frames

• Stretcher matrices

• Why A.B is unlikely to be the same as B.A given 2D matrices A and B

• Inverse matrices

**Lesson 4: Making 2D Matrices**

• Using two perpendicular frames and two stretch factors to make matrices whose hits have desired outcomes

• Inverting matrices made this way

• Making matrices whose hits:

o stretch along a given perpendicular frame

o reflect about a given line

o project onto a given line

• Ray tracing

• Parabolic, spherical, elliptic and hyperbolic reflectors, stealth technology

**Lesson 5: SVD Analysis of 2D Matrices**

• The SVD (Singular Value Decomposition) says that corresponding to any 2D matrix A are two perpendicular frames and two stretch factors that can be used to duplicate A

• Using SVD stretch factors to recognize invertible matrices and then invert them

• The determinant of a 2D matrix in terms of the SVD stretch factors

• Why the determinant of Inverse[A] is the inverse of the determinant of A

• Rank of a 2D matrix

• Using 2D matrices to solve systems of linear equations

• Eigenvalues and eigenvectors of 2D matrices

• Hand calculations involving Cramer’s rule and Gaussian elimination (optional)

**Lesson 6: 3D Matrices**

This lesson repeats the ideas of Lessons 3,4, and 5 in 3D

**Lesson 7: Beyond 3D**

• The SVD (Singular Value Decomposition) says that corresponding to any arbitrary matrix A (square or non‐square) are two perpendicular frames and a list of stretch factors that can be used to duplicate A

• Rank of a matrix in terms of the SVD stretch factors

• The meaning of full rank

• Recognizing when a given system of n linear equations in k unknowns has

a. exactly one solution (exactly determined)

b. many solutions (under determined)

c. no solution (over determined)

• How to find find solutions of linear systems when they exist

• Using SVD to explicitly construct the PseudoInverse for getting best least squares solutions to over determined systems of linear equations

**Lesson 8: Roundoff (Optional)**

• Creative rounding of matrices via the Singular Value Decomposition and image compression

• Principal Component Analysis (PCA) of data via the Singular Value Decomposition

• Ill‐conditioned matrices: The trouble ill‐conditioned matrices can cause and how to use the Singular Value Decomposition to recognize them.

**Lesson 9: Subspaces**

• Every set of vectors in nD spans a subspace of nD

• Projecting onto a subspace of nD

• Calculating the dimension of a subspace of nD

• A set of k vectors in nD is linearly independent if it spans a k‐dimensional subspace of nD

• Traditional definitions of linear independence

• Orthonormal sets

• Gram Schmidt process

• Alien plots coming from projections of highD surfaces onto 3D subspaces

• Perpendicular complement of a subspace

• Null spaces of matrices.

**Lesson 10: Eigensense: Diagonalizable Matrices, Matrix Exponential, Matrix Powers and Dynamical Systems**

• Eigenvalues, eigenvectors, and using them to recognize diagonalizable matrices

• Complex eigenvalues and eigenvectors

• The matrix exponential for diagonalizable and non-diagonalizable matrices

• Eigenvalues reveal long‐term behavior of matrix exponentials and matrix powers

• Using matrix exponentials and matrix powers to solve continuous dynamical systems (systems of linear differential equations) and discrete dynamical systems (systems of difference equations)

**Lesson 11: The Spectral Theorem for Symmetric Matrices and the Holy Grail of Matrix Theory**

• This is the main theoretical lesson. Discussion of the spectral theorem and its proof. Given an arbitrary matrix A, using the spectral theorem applied to Transpose[A].A to explain why every matrix has a singular value decomposition.

• Using an orthonormal basis of eigenvectors of Transpose[A].A to read off:

o an orthonormal basis of the column space R[A] of the matrix A

o an orthonormal basis of the null space N[A] of the matrix A

o an orthonormal basis of the row space of the matrix A

o a construction of the PseudoInverse of the matrix A

• Positive definite and positive semidefinite matrices

• Quadratic forms

• Grammian matrices

**Lesson 12: Function Spaces**

• Functions as vectors

• The root‐mean‐square distance between two functions on an interval

• Weighted root‐mean‐square distance

• The dot product of two functions

• The component of one function in the direction of another

• Orthogonal sets of functions: Sine systems, Cosine systems, Sin‐Cosine systems, Legendre Polynomial system

• Sets of functions orthogonal with respect to a weight function

• Chebyshev polynomials

• Gram‐Schmidt process

• Fourier approximation and orthogonal functions

• Fourier Sine approximation and the heat and wave equations

• Using Fourier methods to bring the Dirac Delta function to life.