 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 22434 students have viewed full stepbystep answer. Linear Algebra was written by and is associated to the ISBN: 9780130084514.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

lAII = 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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Multiplier eij.
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).

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

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.

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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!