Linear Algebra Study Guide



You will use it from college all the way to graduate school and beyond. FREE chapters on Linear equations, Determinant, and more in the trial version.


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
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
Eigenvalue, eigenvector and eigenspace
Cayley-Hamilton theorem
Spread of a matrix
Symbolic Computation of Matrix Eigenvalues
Jordan normal form
Matrix inversion,
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
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
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
Normal Vector
Tangential and Normal Components
Scalar multiplication
Linear combination
Linear span
Linear independence
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
Orthogonal complement
Orthogonal projection

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