Linear Algebra II
Structure theorems and the geometry of linear maps.
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Diagonalization and the Spectral Theorem
Similarity, symmetric matrices, and orthogonal diagonalization.
Inner Product Spaces
Abstract inner products, Gram-Schmidt, and orthogonal complements.
Singular Value Decomposition and Applications
SVD, low-rank approximation, and least squares revisited.
Abstract Vector Spaces and Linear Maps
General fields, quotient constructions, and change of basis in full generality.