 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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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.)

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).

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.

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

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

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

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).

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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.

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.