#math/linearalgebra #moc # Linear Algebra Done Right by Sheldon Axler ## 0. Knowledge and Notation * [[1A1 - Complex Numbers]] * [[1A2 - Fields]] - [[1A3 - Problem Set]] ## 1. Vector Spaces * [[1B1 - Definition of Vector Space]] - [[1B2 - Problem Set]] * [[1C1 - Subspaces]] * [[1C2 - Sums and Direct Sums of Subspaces]] ## 2. Finite-Dimensional Vector Spaces * [[Span and Linear Independence]] * [[Basis - The Minimal Generator]] * [[Dimension - The Invariant Integer]] ## 3. Linear Maps * [[The Vector Space of Linear Maps]] * [[Null Space and Range]] * [[Fundamental Theorem of Linear Maps]] * [[Matrices - Representations of Maps]] * [[Invertibility and Isomorphisms]] ## 4. Polynomials * [[Roots of Polynomials]] * [[Division Algorithm for Polynomials]] ## 5. Eigenvalues and Eigenvectors * [[Invariant Subspaces]] * [[Eigenvalues and Eigenvectors - Definitions]] * [[The Existence of Eigenvalues over C]] * [[Upper-Triangular Matrices]] * [[Diagonalizable Operators]] ## 6. Inner Product Spaces Adding geometry to the algebra. * [[Inner Products and Norms]] * [[Orthonormal Bases]] * [[Gram-Schmidt Procedure]] * [[Orthogonal Complements and Projections]] ## 7. Operators on Inner Product Spaces TODO * [[Self-Adjoint and Normal Operators]] * [[The Spectral Theorem]] * [[Isometries and Polar Decomposition]] * [[Singular Value Decomposition (SVD)]] ## 8. Operators on Complex Vector Spaces TODO * [[Generalized Eigenvectors]] * [[Characteristic Polynomials]] * [[Jordan Form]] ## 9. Operators on Real Vector Spaces TODO * [[Operators on Real Inner Product Spaces]] * [[The Real Spectral Theorem]] ## 10. Trace and Determinant TODO * [[Trace - The Sum of Eigenvalues]] * [[Determinant - The Product of Eigenvalues]]