The following video provides an outline of all the topics you would expect to see in a typical **Linear Algebra** class (i.e., Matrix Algebra).

All of the topics are covered in detail in our Online Linear Algebra Course. The course contains full lectures, detailed examples, and also includes practice final exams.

## Linear Algebra Overview

Linear Algebra Overview

## Linear Equations in Linear Algebra

- Systems of Linear Equations
- Row Reduction and Echelon Forms
- Vector Equations
- The Matrix Equation Ax=b
- Solution Sets of Linear Systems
- Linear Independence
- Linear Transformations and the Matrix of a Linear Transformation
- Applications of Linear Systems and Linear Models in Business, Science and Engineering

## Matrix Algebra

- Matrix Operations and Determinants
- The Inverse of a Matrix
- Characterizations of Invertible Matrices
- Partitioned Matrices
- Matrix Factorization (LU Decomposition)
- Applications to Computer Graphics

## Determinants

- Properties to Determinants
- Cramer’s Rule, Volume, and Linear Transformations

## Vector Spaces

- Vector Spaces and Subspaces
- Null Spaces, Column Spaces, Row Spaces and Linear Transformations
- Linearly Independent Sets; Bases
- Coordinate Systems
- The Dimension of a Vector Space
- Rank
- Change of Basis
- Applications to Markov Chains

## Eigenvalues and Eigenvectors

- Eigenvectors and Eigenvalues
- The Characteristic Equation
- Diagonolization
- Eigenvectors and Linear Transformations
- Complex Eigenvalues

## Orthogonality and Least Squares

- Inner Product, Length, and Orthogonality
- Orthogonal Sets
- Orthogonal Projections
- The Gram-Schmidt Process and QR Factorization
- Least-Squares Problems
- Applications to Linear Models (Line of Regressions)
- Inner Product Spaces

## Symmetric Matrices and Quadratic Forms

- Diagonalization of Symmetric Matrices
- Quadratic Forms

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