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Textbooks / Math / College Algebra 7

College Algebra 7th Edition - Solutions by Chapter

College Algebra | 7th Edition | ISBN: 9781439048610 | Authors: Richard N. Aufmann, Vernon C. Barker, Richard D. Nation

Full solutions for College Algebra | 7th Edition

ISBN: 9781439048610

College Algebra | 7th Edition | ISBN: 9781439048610 | Authors: Richard N. Aufmann, Vernon C. Barker, Richard D. Nation

College Algebra | 7th Edition - 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
  • 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.

  • 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!

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • 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].

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • Gauss-Jordan method.

    Invert A by row operations on [A I] to reach [I A-I].

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • 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.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • 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.

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • 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).

  • 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.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • 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 ().

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • 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.

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