 Chapter 1: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
 Chapter 1.1: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
 Chapter 1.2: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
 Chapter 1.3: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
 Chapter 1.4: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
 Chapter 1.5: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
 Chapter 1.6: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
 Chapter 1.7: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
 Chapter 2: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 2.1: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 2.2: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 2.3: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 2.4: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 2.5: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 2.6: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 2.7: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 2.8: MATRICES AND LINEAR TRANSFORMATIONS
 Chapter 3: DETERMINANTS
 Chapter 3.1: DETERMINANTS
 Chapter 3.2: DETERMINANTS
 Chapter 4: DETERMINANTS
 Chapter 4.1: SUBSPACES AND THEIR PROPERTIES
 Chapter 4.2: SUBSPACES AND THEIR PROPERTIES
 Chapter 4.3: SUBSPACES AND THEIR PROPERTIES
 Chapter 4.4: DETERMINANTS
 Chapter 4.5: DETERMINANTS
 Chapter 5: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION
 Chapter 5.1: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION
 Chapter 5.2: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION
 Chapter 5.3: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION
 Chapter 5.4: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION
 Chapter 5.5: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION
 Chapter 6.1: ORTHOGONALITY
 Chapter 6.2: ORTHOGONALITY
Elementary Linear Algebra: A Matrix Approach 2nd Edition  Solutions by Chapter
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Elementary Linear Algebra: A Matrix Approach  2nd Edition  Solutions by Chapter
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Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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

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.

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).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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 one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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