- Chapter 1: AN INTRODUCTION TO DATA AND FUNCTIONS
- Chapter 2: RATES OF CHANGE AND LINEAR FUNCTIONS
- Chapter 3: WHEN LINES MEET: LINEAR SYSTEMS
- Chapter 4: THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE
- Chapter 5: GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS
- Chapter 6: LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS
- Chapter 7: POWER FUNCTIONS
- Chapter 8: QUADRATICS AND THE MATHEMATICS OF MOTION
- Chapter 9: NEW FUNCTIONS FROM OLD
Explorations in College Algebra 5th Edition - Solutions by Chapter
Full solutions for Explorations in College Algebra | 5th Edition
ISBN: 9780470466445
Solutions by Chapter
Explorations in College Algebra was written by and is associated to the ISBN: 9780470466445. This expansive textbook survival guide covers the following chapters: 9. This textbook survival guide was created for the textbook: Explorations in College Algebra, edition: 5. Since problems from 9 chapters in Explorations in College Algebra have been answered, more than 13252 students have viewed full step-by-step answer. The full step-by-step solution to problem in Explorations in College Algebra were answered by , our top Math solution expert on 12/23/17, 04:55PM.
Key Math Terms and definitions covered in this textbook
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
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Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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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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Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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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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Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
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Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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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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Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.
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Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
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Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.