×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
Textbooks / Math / Geometry (Holt McDougal Larson Geometry) 1

Geometry (Holt McDougal Larson Geometry) 1st Edition - Solutions by Chapter

Geometry (Holt McDougal Larson Geometry) | 1st Edition | ISBN: 9780618595402 | Authors: Ron Larson Laurie Boswell Timothy D. Kanold, Lee Stiff

Full solutions for Geometry (Holt McDougal Larson Geometry) | 1st Edition

ISBN: 9780618595402

Geometry (Holt McDougal Larson Geometry) | 1st Edition | ISBN: 9780618595402 | Authors: Ron Larson Laurie Boswell Timothy D. Kanold, Lee Stiff

Geometry (Holt McDougal Larson Geometry) | 1st Edition - Solutions by Chapter

The full step-by-step solution to problem in Geometry (Holt McDougal Larson Geometry) were answered by , our top Math solution expert on 03/02/18, 04:34PM. This expansive textbook survival guide covers the following chapters: 12. Geometry (Holt McDougal Larson Geometry) was written by and is associated to the ISBN: 9780618595402. This textbook survival guide was created for the textbook: Geometry (Holt McDougal Larson Geometry), edition: 1. Since problems from 12 chapters in Geometry (Holt McDougal Larson Geometry) have been answered, more than 22138 students have viewed full step-by-step answer.

Key Math Terms and definitions covered in this textbook
  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • 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].

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Determinant IAI = det(A).

    Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > 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).

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
×
Reset your password