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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