 Chapter 1.1: Vectors
 Chapter 1.2: Dot Product
 Chapter 1.3: Hyperplanes in Rn
 Chapter 1.4: Systems of Linear Equations and Gaussian Elimination
 Chapter 1.5: The Theory of Linear Systems
 Chapter 1.6: Some Applications
 Chapter 2.1: Matrix Operations
 Chapter 2.2: Linear Transformations: An Introduction
 Chapter 2.3: Inverse Matrices
 Chapter 2.4: Elementary Matrices: Rows Get Equal Time
 Chapter 2.5: The Transpose
 Chapter 3.1: Subspaces of Rn
 Chapter 3.2: The Four Fundamental Subspaces
 Chapter 3.3: Linear Independence and Basis
 Chapter 3.4: Dimension and Its Consequences
 Chapter 3.5: A Graphic Example
 Chapter 3.6: AbstractVector Spaces
 Chapter 4.1: Inconsistent Systems and Projection
 Chapter 4.2: Orthogonal Bases
 Chapter 4.3: The Matrix of a Linear Transformation and the ChangeofBasis Formula
 Chapter 4.4: Linear Transformations on Abstract Vector Spaces
 Chapter 5.1: Properties of Determinants
 Chapter 5.2: Cofactors and Cramers Rule
 Chapter 5.3: Signed Area in R2 and SignedVolume in R3
 Chapter 6.1: The Characteristic Polynomial
 Chapter 6.2: Diagonalizability
 Chapter 6.3: Applications
 Chapter 6.4: The Spectral Theorem
 Chapter 7.1: Complex Eigenvalues and Jordan Canonical Form
 Chapter 7.2: Computer Graphics and Geometry
 Chapter 7.3: Matrix Exponentials and Differential Equations
Linear Algebra: A Geometric Approach 2nd Edition  Solutions by Chapter
Full solutions for Linear Algebra: A Geometric Approach  2nd Edition
ISBN: 9781429215213
Linear Algebra: A Geometric Approach  2nd Edition  Solutions by Chapter
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra: A Geometric Approach, edition: 2. This expansive textbook survival guide covers the following chapters: 31. The full stepbystep solution to problem in Linear Algebra: A Geometric Approach were answered by , our top Math solution expert on 03/15/18, 05:30PM. Linear Algebra: A Geometric Approach was written by and is associated to the ISBN: 9781429215213. Since problems from 31 chapters in Linear Algebra: A Geometric Approach have been answered, more than 3029 students have viewed full stepbystep answer.

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).

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.

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.

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.

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

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

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.

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

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 columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).