Matrix Theory and Linear Algebra

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https://www.mathstat.dal.ca/~selinger/linear-algebra/ Contents Preface Systems of linear equations Geometric view of systems of equations Algebraic view of systems of equations Elementary operations Gaussian elimination Gauss-Jordan elimination Homogeneous systems Uniqueness of the reduced echelon form Fields Application: Balancing chemical reactions Application: Dimensionless variables Application: Resistor networks Vectors in Rn Points and vectors Addition Scalar multiplication Linear combinations Length of a vector The dot product Definition and properties The Cauchy-Schwarz and triangle inequalities The geometric significance of the dot product Orthogonal vectors Projections The cross product Right-handed systems of vectors Geometric description of the cross product Algebraic definition of the cross product The box product Lines and planes in Rn Lines Planes Matrices Definition and equality Addition Scalar multiplication Matrix multiplication Multiplying a matrix and a vector Matrix multiplication Properties of matrix multiplication Matrix inverses Definition and uniqueness Computing inverses Using the inverse to solve a system of equations Properties of the inverse Right and left inverses Elementary matrices Elementary matrices and row operations Inverses of elementary matrices Elementary matrices and reduced echelon forms Writing an invertible matrix as a product of elementary matrices More properties of inverses The transpose Matrix arithmetic modulo p Application: Cryptography Spans, linear independence, and bases in Rn Spans Linear independence Redundant vectors and linear independence The casting-out algorithm Alternative characterization of linear independence Properties of linear independence Linear independence and linear combinations Removing redundant vectors Subspaces of Rn Basis and dimension Definition of basis Examples of bases Bases and coordinate systems Dimension More properties of bases and dimension Column space, row space, and null space of a matrix Linear transformations in Rn Linear transformations The matrix of a linear transformation Geometric interpretation of linear transformations Properties of linear transformations Application: Perspective rendering Determinants Determinants of 2x2- and 3x3-matrices Minors and cofactors The determinant of a triangular matrix Determinants and row operations Properties of determinants Application: A formula for the inverse of a matrix Application: Cramer's rule Eigenvalues, eigenvectors, and diagonalization Eigenvectors and eigenvalues Finding eigenvalues Geometric interpretation of eigenvectors Diagonalization Application: Matrix powers Application: Solving recurrences Application: Systems of linear differential equations Differential equations Systems of linear differential equations Example: coupled train cars Application: The matrix exponential Properties of eigenvectors and eigenvalues The Cayley-Hamilton Theorem Complex eigenvalues and eigenvectors Vector spaces Definition of vector spaces Linear combinations, span, and linear independence Subspaces Basis and dimension Application: Error correcting codes Linear transformation of vector spaces Definition and examples The algebra of linear transformations Linear transformations defined on a basis The matrix of a linear transformation Inner product spaces Real inner product spaces Orthogonality The Gram-Schmidt orthogonalization procedure Orthogonal projections and Fourier series Application: Least squares approximations and curve fitting Orthogonal functions and orthogonal matrices Diagonalization of symmetric matrices Positive semidefinite and positive definite matrices Application: Simplification of quadratic forms Complex inner product spaces Unitary and hermitian matrices Application: Principal component analysis Complex numbers The complex numbers Geometric interpretation The fundamental theorem of algebra Answers to selected exercises Index