Linear Algebra Study Guide

Boost Your grades with this illustrated Study Guide. You will use it from college all the way to graduate school and beyond.

Features

Clear and concise explanations
Difficult concepts are explained in simple terms
Illustrated with graphs and diagrams


Table of Contents

I. Linear equations

System of linear equations
Determinant

Minor
Cauchy-Binet formula


Cramer's rule
Gaussian elimination
Gauss-Jordan elimination
Strassen algorithm


II. Matrices

Matrix addition
Matrix multiplication
Basis transformation matrix
Characteristic polynomial, Characteristic Equation
Trace
Eigenvalue, eigenvector and eigenspace

Cayley-Hamilton theorem
Spread of a matrix
Symbolic Computation of Matrix Eigenvalues


Jordan normal form
Rank
Matrix inversion,

Pseudoinverse


Adjugate
Transpose

Dot product
Symmetric matrix
Matrix congruence

Congruence relation


Orthogonal matrix
Skew-symmetric matrix
Conjugate transpose

Unitary matrix
Hermitian matrix, Antihermitian




Positive definite: matrix, function, bilinear form
Identity matrix
Pfaffian
Projection
Diagonal matrix, main diagonal

Diagonalizable matrix


Similar matrix
Tridiagonal matrix
Hessenberg matrix
Triangular matrix
Spectral theorem
Stochastic matrix
Toeplitz matrix

Circulant matrix


Hankel matrix
Vandermonde matrix
Block matrix
(0,1)-matrix
Normal Matrix
Sparse matrix
Woodbury matrix identity
Perron-Frobenius theorem
List of Matrices


III. Matrix decompositions

Block LU Decomposition
Cholesky decomposition
LU decomposition
QR decomposition
Spectral theorem
Singular value decomposition
Schur decomposition

Schur complement




IV. Computations

Transformation Matrix
Householder transformation
Least squares, linear least squares
Gram-Schmidt process


V. Vectors

Unit Vector
Pseudovector
Normal Vector
Tangential and Normal Components
Scalar multiplication
Linear combination
Linear span
Linear independence
Basis


VI. Vector spaces

Basis=Hamel basis
Dimension theorem for vector spaces=Hamel dimension
Examples of vector spaces
Linear map

Galilean transformation, Lorentz transformation


Row and Column space
Null space
Rank-nullity theorem
Dual space

Linear function
Linear functional


Orthogonality
Orthogonal complement
Orthogonal projection


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