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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 5330 students have viewed full step-by-step answer.
Key Math Terms and definitions covered in this textbook
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Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
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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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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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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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Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
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lA-II = 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.
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Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.
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Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
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Pseudoinverse A+ (Moore-Penrose 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).
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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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.
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Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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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 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.