 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 3884 students have viewed full stepbystep answer.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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).

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.

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.

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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.