- 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 24165 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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Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
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Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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Column space C (A) =
space of all combinations of the columns of A.
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Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } 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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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
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Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= 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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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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Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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Rotation matrix
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
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Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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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.