 Chapter 1.1: Introduction
 Chapter 1.2: Vector Spaces
 Chapter 1.3: Subspaces
 Chapter 1.4: Linear Combinations and Systems of Linear Equations
 Chapter 1.5: Linear Dependence and Linear Independence
 Chapter 1.6: Bases and Dimension
 Chapter 1.7: Maximal Linearly Independent Subsets
 Chapter 2.1: Linear Transformations, Null Spaces, and Ranges
 Chapter 2.2: The Matrix Representation of a Linear Transformation
 Chapter 2.3: Composition of Linear Transformations and Matrix Multiplication
 Chapter 2.4: Invertibility and Isomorphisms
 Chapter 2.5: The Change of Coordinate Matrix
 Chapter 2.6: Dual Spaces
 Chapter 2.7: Homogeneous Linear Differential Equations with Constant Coefficients
 Chapter 3.1: Elementary Matrix Operations and Elementary Matrices
 Chapter 3.2: The Rank of a Matrix and Matrix Inverses
 Chapter 3.3: Systems of Linear EquationsTheoretical Aspects
 Chapter 3.4: Systems of Linear EquationsComputational Aspects
 Chapter 4.1: Determinants of Order 2
 Chapter 4.2: Determinants of Order //
 Chapter 4.3: Properties of Determinants
 Chapter 4.4: SummaryImportant Facts about Determinants
 Chapter 4.5: A Characterization of the Determinant
 Chapter 5.1: Eigenvalues and Eigenvectors
 Chapter 5.2: Diagonalizability
 Chapter 5.3: Matrix Limits and Markov Chains
 Chapter 5.4: Invariant Subspaces and the CayleyHamilton Theorem
 Chapter 6.1: Inner Products and Norms
 Chapter 6.10:
 Chapter 6.11: The Geometry of Orthogonal Operators
 Chapter 6.2: GramSchmidt Orthogonalization Process
 Chapter 6.3: The Adjoint of a Linear Operator
 Chapter 6.4: Normal and SelfAdjoint Operators
 Chapter 6.5: Unitary and Orthogonal Operators and Their Matrices
 Chapter 6.6: Orthogonal Projections and the Spectral Theorem
 Chapter 6.7: The Singular Value Decomposition and the Pseudoinverse
 Chapter 6.8: Bilinear and Quadratic Forms
 Chapter 6.9: Einstein's Special Theory of Relativity
 Chapter 7.1: The Jordan Canonical Form I
 Chapter 7.2: The Jordan Canonical Form II
 Chapter 7.3: The Minimal Polynomial
 Chapter 7.4: The Rational Canonical Form
 Chapter `6.10:
Linear Algebra 4th Edition  Solutions by Chapter
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Linear Algebra  4th Edition  Solutions by Chapter
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra , edition: 4. This expansive textbook survival guide covers the following chapters: 43. The full stepbystep solution to problem in Linear Algebra were answered by , our top Math solution expert on 07/25/17, 09:33AM. Since problems from 43 chapters in Linear Algebra have been answered, more than 9910 students have viewed full stepbystep answer. Linear Algebra was written by and is associated to the ISBN: 9780130084514.

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).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.