 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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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

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.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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.

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.

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.

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.