- Chapter 1: Matrices and Systems of Equations
- Chapter 1.1: Systems of Linear Equations
- Chapter 1.2: Row Echelon Form
- Chapter 1.3: Matrix Arithmetic
- Chapter 1.4: Matrix Algebra
- Chapter 1.5: Elementary Matrices
- Chapter 1.6: Partitioned Matrices
- Chapter 2: Determinants
- Chapter 2.1: The Determinant of a Matrix
- Chapter 2.2: Properties of Determinants
- Chapter 2.3: Additional Topics and Applications
- Chapter 3: Vector Spaces
- Chapter 3.1: Definition and Examples
- Chapter 3.2: Subspaces
- Chapter 3.3: Linear Independence
- Chapter 3.4: Basis and Dimension
- Chapter 3.5: Change of Basis
- Chapter 3.6: Row Space and Column Space
- Chapter 4: Linear Transformations
- Chapter 4.1: Definition and Examples
- Chapter 4.2: Matrix Representations of Linear Transformations
- Chapter 4.3: Similarity
- Chapter 5: Orthogonality
- Chapter 5.1: The Scalar Product in Rn
- Chapter 5.2: Orthogonal Subspaces
- Chapter 5.3: Least Squares Problems
- Chapter 5.4: Inner Product Spaces
- Chapter 5.5: Orthonormal Sets
- Chapter 5.6: The GramSchmidt Orthogonalization Process
- Chapter 5.7: Orthogonal Polynomials
- Chapter 6: Eigenvalues
- Chapter 6.1: Eigenvalues and Eigenvectors
- Chapter 6.2: Systems of Linear Differential Equations
- Chapter 6.3: Diagonalization
- Chapter 6.4: Hermitian Matrices
- Chapter 6.5: The Singular Value Decomposition
- Chapter 6.6: Quadratic Forms
- Chapter 6.7: Positive Definite Matrices
- Chapter 6.8: Nonnegative Matrices
- Chapter 7: Numerical Linear Algebra
- Chapter 7.1: Floating-Point Numbers
- Chapter 7.2: Gaussian Elimination
- Chapter 7.3: Pivoting Strategies
- Chapter 7.4: Matrix Norms and Condition Numbers
- Chapter 7.5: Orthogonal Transformations
- Chapter 7.6: The Eigenvalue Problem
- Chapter 7.7: Least Squares Problems
Linear Algebra with Applications 8th Edition - Solutions by Chapter
Full solutions for Linear Algebra with Applications | 8th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Every v in V is orthogonal to every w in W.
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).
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).