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# Linear Algebra

### Overview

This course on Linear Algebra aims to teach students how to solve systems of linear equations using matrices, understand matrix operations and properties, find matrix inverses, determine eigenvalues and eigenvectors, and perform singular value decomposition. The course covers topics such as Gaussian elimination, determinants, orthogonal vectors, and symmetric matrices. The teaching method includes theoretical explanations, proofs, and practical examples. This course is intended for individuals interested in gaining a solid understanding of linear algebra concepts and applications.

### Syllabus

Linear Equations Introduction.
Solving Systems of Linear Equations.
Using Matrices to Solve Systems of Linear Equations.
Reduced Row Echelon Form.
Gaussian Elimination.
Existence and Uniqueness of Solutions.
Linear Equations Setup.
Matrix Multiplication.
Properties of Matrix Multiplication.
Interpretations of Matrix Multiplication.
Introduction to Vectors (Math 210).
Solving Vector Equations.
Solving Matrix Equations.
Matrix Inverses.
Matrix Inverses for 2x2 Matrices.
Equivalent Conditions for a Matrix to be Invertible.
Properties of Matrix Inverses.
Transpose.
Symmetric and Skew-symmetric Matrices.
Trace.
The Determinant of a Matrix.
Determinant and Elementary Row Operations.
Determinant Properties..
Invertible Matrices and Their Determinants.
Eigenvalues and Eigenvectors.
Properties of Eigenvalues.
Diagonalizing Matrices.
Dot Product (Linear Algebra).
Unit Vectors.
Orthogonal Vectors.
Orthogonal Matrices.
Symmetric Matrices and Eigenvectors and Eigenvalues.
Symmetric Matrices and Eigenvalues and Eigenvectors - Proofs.
Diagonalizing Symmetric Matrices.
Linearly Independent Vectors.
Gram-Schmidt Orthogonalization.
Singular Value Decomposition - Introduction.
Singular Value Decomposition - How To Find It.
Singular Value Decomposition - Why It Works.

Linda Green

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