\contentsline {part}{I\hspace {1em}Fundamentals}{1}{}%
\contentsline {chapter}{\numberline {1}Linear equations}{3}{}%
\contentsline {section}{\numberline {1.1}Gaussian elimination}{3}{}%
\contentsline {subsection}{\numberline {1.1.1}Non-unique solutions}{3}{}%
\contentsline {chapter}{\numberline {2}Matrix theory}{5}{}%
\contentsline {section}{\numberline {2.1}Matrixes and Vectors}{5}{}%
\contentsline {section}{\numberline {2.2}Matrix algebra}{6}{}%
\contentsline {subsection}{\numberline {2.2.1}Matrix addition and scalar multiplication}{7}{}%
\contentsline {subsection}{\numberline {2.2.2}Laws of matrix addition and scalar multiplication}{7}{}%
\contentsline {subsection}{\numberline {2.2.3}Matrix multiplication}{8}{}%
\contentsline {subsection}{\numberline {2.2.4}Laws of matrix multiplication}{8}{}%
\contentsline {subsection}{\numberline {2.2.5}Relationship to linear equations}{8}{}%
\contentsline {section}{\numberline {2.3}Determinant}{9}{}%
\contentsline {subsection}{\numberline {2.3.1}Invertible matrix}{9}{}%
\contentsline {subsection}{\numberline {2.3.2}Invertible matrix theorem}{10}{}%
\contentsline {subsection}{\numberline {2.3.3}Properties of the determinant}{10}{}%
\contentsline {subsection}{\numberline {2.3.4}Laplace expansion}{10}{}%
\contentsline {subsection}{\numberline {2.3.5}Leibniz' formula}{10}{}%
\contentsline {subsection}{\numberline {2.3.6}Cramer's rule}{10}{}%
\contentsline {subsection}{\numberline {2.3.7}Principle minors}{10}{}%
\contentsline {subsection}{\numberline {2.3.8}Matrix determinant lemma}{10}{}%
\contentsline {section}{\numberline {2.4}Transposition}{10}{}%
\contentsline {subsection}{\numberline {2.4.1}Symmetric matrix}{11}{}%
\contentsline {subsection}{\numberline {2.4.2}Orthogonal matrix*}{11}{}%
\contentsline {subsection}{\numberline {2.4.3}Hermitian matrix*}{11}{}%
\contentsline {section}{\numberline {2.5}Miscellaneous formulae}{11}{}%
\contentsline {chapter}{\numberline {3}Linear spaces}{13}{}%
\contentsline {section}{\numberline {3.1}Linear space}{13}{}%
\contentsline {subsection}{\numberline {3.1.1}Properties of linear spaces}{14}{}%
\contentsline {subsection}{\numberline {3.1.2}Examples of linear spaces}{15}{}%
\contentsline {section}{\numberline {3.2}Linear subspace}{15}{}%
\contentsline {subsection}{\numberline {3.2.1}Examples of linear subspaces}{15}{}%
\contentsline {section}{\numberline {3.3}Linear combinations}{16}{}%
\contentsline {subsection}{\numberline {3.3.1}Linear span}{16}{}%
\contentsline {subsection}{\numberline {3.3.2}Linear independence}{17}{}%
\contentsline {section}{\numberline {3.4}Basis (Linear space)}{17}{}%
\contentsline {section}{\numberline {3.5}Dimension}{18}{}%
\contentsline {section}{\numberline {3.6}Spanning set theorem}{20}{}%
\contentsline {section}{\numberline {3.7}Linear space constructions}{20}{}%
\contentsline {subsection}{\numberline {3.7.1}Direct sum of linear spaces}{20}{}%
\contentsline {subsection}{\numberline {3.7.2}Direct product of linear spaces}{20}{}%
\contentsline {subsection}{\numberline {3.7.3}Quotient linear space}{20}{}%
\contentsline {chapter}{\numberline {4}Linear maps}{23}{}%
\contentsline {subsection}{\numberline {4.0.1}Kernel (linear maps)}{24}{}%
\contentsline {subsection}{\numberline {4.0.2}Image (linear maps)}{24}{}%
\contentsline {subsection}{\numberline {4.0.3}Exaples of linear maps}{24}{}%
\contentsline {subsection}{\numberline {4.0.4}Linear isomorphisms}{25}{}%
\contentsline {subsection}{\numberline {4.0.5}Linear endomorphisms}{25}{}%
\contentsline {subsection}{\numberline {4.0.6}Linear forms}{25}{}%
\contentsline {section}{\numberline {4.1}Rank-nullity theorem}{25}{}%
\contentsline {chapter}{\numberline {5}Normed linear spaces}{27}{}%
\contentsline {section}{\numberline {5.1}Norm}{27}{}%
\contentsline {subsection}{\numberline {5.1.1}Examples of normed linear spaces}{28}{}%
\contentsline {section}{\numberline {5.2}Inner product spaces}{28}{}%
\contentsline {subsection}{\numberline {5.2.1}Inner product}{28}{}%
\contentsline {subsection}{\numberline {5.2.2}Canonically induced norm}{29}{}%
\contentsline {subsection}{\numberline {5.2.3}Cauchy-Schwarz inequality}{29}{}%
\contentsline {subsection}{\numberline {5.2.4}Pythagorean theorem}{29}{}%
\contentsline {section}{\numberline {5.3}Euclidean space}{30}{}%
\contentsline {subsection}{\numberline {5.3.1}Dot product}{30}{}%
\contentsline {subsection}{\numberline {5.3.2}Results of Euclidean geometry}{31}{}%
\contentsline {subsection}{\numberline {5.3.3}Cross product}{31}{}%
\contentsline {section}{\numberline {5.4}Dual spaces}{32}{}%
\contentsline {subsection}{\numberline {5.4.1}Annihilators}{32}{}%
\contentsline {subsection}{\numberline {5.4.2}Transpose of linear map}{32}{}%
\contentsline {chapter}{\numberline {6}Orthogonality}{33}{}%
\contentsline {section}{\numberline {6.1}Spanning set theorem}{33}{}%
\contentsline {section}{\numberline {6.2}Orthogonal pair of vectors}{33}{}%
\contentsline {section}{\numberline {6.3}Orthonormal set}{33}{}%
\contentsline {section}{\numberline {6.4}Orthogonal complement}{34}{}%
\contentsline {section}{\numberline {6.5}Best approximation theorem}{34}{}%
\contentsline {section}{\numberline {6.6}Orthogonal decomposition theorem}{34}{}%
\contentsline {section}{\numberline {6.7}Vector projection on 1 dimensional linear subspace}{34}{}%
\contentsline {section}{\numberline {6.8}Vector projection on $n$ dimensional linear subspace}{34}{}%
\contentsline {section}{\numberline {6.9}Gram-Schmidt process}{34}{}%
\contentsline {section}{\numberline {6.10}Orthogonal matrix}{35}{}%
\contentsline {chapter}{\numberline {7}Eigenequations}{37}{}%
\contentsline {section}{\numberline {7.1}Eigenequations}{37}{}%
\contentsline {subsection}{\numberline {7.1.1}Solving eigenequations}{37}{}%
\contentsline {subsection}{\numberline {7.1.2}Examples of eigenequations}{38}{}%
\contentsline {subsection}{\numberline {7.1.3}Eigenvalues}{38}{}%
\contentsline {subsection}{\numberline {7.1.4}Eigenvectors}{38}{}%
\contentsline {section}{\numberline {7.2}Eigenspaces}{38}{}%
\contentsline {subsection}{\numberline {7.2.1}Eigenspaces}{38}{}%
\contentsline {subsection}{\numberline {7.2.2}Properties of eigenspaces}{39}{}%
\contentsline {section}{\numberline {7.3}Eigendecomposition and matrix diagonalization}{39}{}%
\contentsline {subsection}{\numberline {7.3.1}Matrix diagonalization}{39}{}%
\contentsline {subsection}{\numberline {7.3.2}OrthMatrix diagonalization}{40}{}%
\contentsline {subsection}{\numberline {7.3.3}Properties of diagonalizable matrixes}{40}{}%
\contentsline {subsection}{\numberline {7.3.4}Eigendecomposition}{40}{}%
\contentsline {chapter}{\numberline {8}Quadratic forms}{41}{}%
\contentsline {part}{II\hspace {1em}Advanced}{43}{}%
\contentsline {chapter}{\numberline {9}Advanced matrix algebra}{45}{}%
\contentsline {section}{\numberline {9.1}Matrix exponential}{45}{}%
\contentsline {section}{\numberline {9.2}Cayley-Hamilton theorem}{46}{}%
\contentsline {chapter}{\numberline {10}Multilinear algebra}{47}{}%
\contentsline {section}{\numberline {10.1}Multilinear maps}{47}{}%
\contentsline {subsection}{\numberline {10.1.1}Outer product}{48}{}%
\contentsline {subsection}{\numberline {10.1.2}Hadamard product}{48}{}%
\contentsline {subsection}{\numberline {10.1.3}Kronecker product}{48}{}%
\contentsline {section}{\numberline {10.2}Tensor product}{48}{}%
\contentsline {section}{\numberline {10.3}Grassmann algebra}{48}{}%
\contentsline {chapter}{\numberline {11}Tensors}{49}{}%
\contentsline {section}{\numberline {11.1}Vectors and covectors (linear forms)}{49}{}%
\contentsline {section}{\numberline {11.2}Ricci calculus}{50}{}%
\contentsline {subsection}{\numberline {11.2.1}Indexing laws}{50}{}%
\contentsline {subsection}{\numberline {11.2.2}Einstein notation}{50}{}%
\contentsline {section}{\numberline {11.3}Tensor}{50}{}%