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College Algebra and Trigonometry 7th Edition - Solutions by Chapter
Full solutions for College Algebra and Trigonometry | 7th Edition
ISBN: 9781439048603
Solutions by Chapter
This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7. The full step-by-step solution to problem in College Algebra and Trigonometry were answered by , our top Math solution expert on 11/15/17, 04:29PM. This expansive textbook survival guide covers the following chapters: 12. College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. Since problems from 12 chapters in College Algebra and Trigonometry have been answered, more than 7195 students have viewed full step-by-step answer.
Key Math Terms and definitions covered in this textbook
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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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].
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Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
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Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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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.
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Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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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.
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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).
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Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.
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Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
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Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.