 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 8022 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).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Fundamental Theorem.
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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Iterative method.
A sequence of steps intended to approach the desired solution.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·