- Chapter 1: Equations and Inequalities
- Chapter 2: Functions and Graphs
- Chapter 3: Polynomial and Rational Functions
- Chapter 4: Exponential and Logarithmic Functions
- Chapter 5: Topics in Analytic Geometry
- Chapter 6: Systems of Equations and Inequalities
- Chapter 7: Matrices
- Chapter 8: Sequences, Series and Probability
- Chapter P: Preliminary Concepts
College Algebra 7th Edition - Solutions by Chapter
Full solutions for College Algebra | 7th Edition
ISBN: 9781439048610
Solutions by Chapter
The full step-by-step solution to problem in College Algebra were answered by , our top Math solution expert on 01/02/18, 08:47PM. This textbook survival guide was created for the textbook: College Algebra, edition: 7. Since problems from 9 chapters in College Algebra have been answered, more than 128016 students have viewed full step-by-step answer. This expansive textbook survival guide covers the following chapters: 9. College Algebra was written by and is associated to the ISBN: 9781439048610.
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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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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Diagonalization
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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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.
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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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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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Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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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.
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Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
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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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Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.