 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: FloatingPoint 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 9th Edition  Solutions by Chapter
Full solutions for Linear Algebra with Applications  9th Edition
ISBN: 9780321962218
Linear Algebra with Applications  9th Edition  Solutions by Chapter
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. This expansive textbook survival guide covers the following chapters: 47. Since problems from 47 chapters in Linear Algebra with Applications have been answered, more than 2494 students have viewed full stepbystep answer. The full stepbystep solution to problem in Linear Algebra with Applications were answered by , our top Math solution expert on 03/15/18, 05:26PM.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
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).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.
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