This course is all about matrices. Topics covered include matrices and their algebra, Gaussian elimination and the LU decomposition, vector spaces, determinants, and the eigenvalue problem.

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Description

Content

This book and accompanying YouTube video lectures is all about matrices, and concisely covers the linear algebra that an engineer should know. We define matrices and how to add and multiply them, and introduce some special types of matrices. We describe the Gaussian elimination algorithm used to solve systems of linear equations and the corresponding LU decomposition of a matrix. We explain the concept of vector spaces and define the main vocabulary of linear algebra. Finally, we develop the theory of determinants and use it to solve the eigenvalue problem.

- Preface
- Week I: Matrices

- Definition of a matrix
- Addition and multiplication of matrices
- Special matrices
- Transpose matrix
- Inner and outer products
- Inverse matrix
- Orthogonal matrices
- Orthogonal matrices example
- Permutation matrices

- Week II: Systems of linear equations

- Gaussian elimination
- Reduced row echelon form
- Computing inverses
- Elementary matrices
- LU decomposition
- Solving (LU)x = b

- Week III: Vector space

- Vector spaces
- Linear independence
- Span, basis and dimension
- Gram-Schmidt process
- Gram-Schmidt process example
- Null space
- Application of the null space
- Column space
- Row space, left null space and rank
- Orthogonal projections
- The least-squares problem
- Solution of the least-squares problem

- Week IV: Eigenvalues and eigenvectors

- Two-by-two and three-by-three determinants
- Laplace expansion
- Leibniz formula
- Properties of a determinant
- The eigenvalue problem
- Finding eigenvalues and eigenvectors (1)
- Finding eigenvalues and eigenvectors (2)
- Matrix diagonalization
- Matrix diagonalization example
- Powers of a matrix
- Powers of a matrix example

- Appendix A Problem solutions