- Chapter 1: Linear Equations
- Chapter 1.1: Introduction to Linear Systems
- Chapter 1.2: Matrices, Vectors, and Gauss-Jordan Elimination
- Chapter 1.3: On the Solutions of Linear Systems; Matrix Algebra
- Chapter 2: Linear Transformations
- Chapter 2.1: Introduction to Linear Transformations and Their Inverses
- Chapter 2.2: Linear Transformations in Geometry
- Chapter 2.3: Matrix Products
- Chapter 2.4: The Inverse of a Linear Transformation
- Chapter 3.1: Image and Kernel of a Linear Transformation
- Chapter 3.2: Subspaces of R"; Bases and Linear Independence
- Chapter 3.3: The Dimension of a Subspace of R"
- Chapter 3.4: Coordinates
- Chapter 4: Linear Spaces
- Chapter 4.1: Introduction to Linear Spaces
- Chapter 4.2: Linear Transformations and Isomorphisms
- Chapter 4.3: Th e Matrix of a Linear Transformation
- Chapter 5: Orthogonality and Least Squares
- Chapter 5.1: Orthogonal Projections and Orthonormal Bases
- Chapter 5.2: Gram-Schmidt Process and QR Factorization
- Chapter 5.3: Orthogonal Transformations and Orthogonal Matrices
- Chapter 5.4: Least Squares and Data Fitting
- Chapter 5.5: Inner Product Spaces
- Chapter 6: Determinants
- Chapter 6.1: Introduction to Determinants
- Chapter 6.2: Properties of the Determinant
- Chapter 6.3: Geometrical Interpretations of the Determinant; Cramers Rule
- Chapter 7: Eigenvalues and Eigenvectors
- Chapter 7.1: Dynamical Systems and Eigenvectors: An Introductory Example
- Chapter 7.2: Finding the Eigenvalues of a Matrix
- Chapter 7.3: Finding the Eigenvectors of a Matrix
- Chapter 7.4: Diagonalization
- Chapter 7.5: Complex Eigenvalues
- Chapter 7.6: Stability
- Chapter 8: Symmetric Matrices and Quadratic Forms
- Chapter 8.1: Symmetric Matrices
- Chapter 8.2: Quadratic Forms
- Chapter 8.3: Singular Values
- Chapter 9.1: An Introduction to Continuous Dynamical Systems
- Chapter 9.2: The Complex Case: Eulers Formula
- Chapter 9.3: Linear Differential Operators and Linear Differential Equations
Linear Algebra with Applications 4th Edition - Solutions by Chapter
Full solutions for Linear Algebra with Applications | 4th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.