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In this course, you will learn:
- Solving linear systems and Gauss-Jordan elimination are two examples of operations on a single matrix.
- Matrix multiplication and elimination matrices are examples of operations on two matrices.
- Vectorization of matrices, including linear combinations and span, linear independence, and subspaces.
- Dot and cross products, as well as the Cauchy-Schwarz and vector triangle inequalities.
- Ax=b and matrix-vector products, including the null and column spaces.
- Transformations, including linear transformations, projections, and transformation composition.
- Inverses, comprising invertible and singular matrices, as well as the solution of systems using inverse matrices.
- Upper and lower triangular matrices, as well as Cramer's rule, are determinants.
- Transposes, including their determinants, as well as the transpose's null (left null) and column (row) spaces.
- Orthogonality and basis change include orthogonal complements, projections into a subspace, least squares, and basis change.
- Orthonormal bases and Gram-Schmidt, including the definition of the orthonormal basis and the Gram-Schmidt technique for converting to an orthonormal basis.
- Eigenvalues and Eigenvectors, includes the determination of eigenvalues and their associated eigenvectors and eigenspaces, as well as eigen in three dimensions.
Syllabus:
- Operations on one matrix
- Operations on two matrices
- Matrices as vectors
- Dot products and cross products
- Matrix-vector products
- Transformations
- Inverses
- Determinants
- Transposes
- Orthogonality and change of basis
- Orthonormal bases and Gram-Schmidt
- Eigenvalues and Eigenvectors
