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Solutions for Chapter 5-3: Medians and Altitudes of Triangles

Full solutions for Geometry | 1st Edition

ISBN: 9780030923456

Solutions for Chapter 5-3: Medians and Altitudes of Triangles

Solutions for Chapter 5-3
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Textbook: Geometry
Edition: 1
Author: Rinehart, Winston Holt
ISBN: 9780030923456

Geometry was written by and is associated to the ISBN: 9780030923456. This textbook survival guide was created for the textbook: Geometry, edition: 1. Chapter 5-3: Medians and Altitudes of Triangles includes 51 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 51 problems in chapter 5-3: Medians and Altitudes of Triangles have been answered, more than 44018 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • 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).

  • Condition number

    cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

  • Distributive Law

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

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

  • Orthogonal subspaces.

    Every v in V is orthogonal to every w in W.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Row space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • 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.

  • Spanning set.

    Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

  • Spectral Theorem A = QAQT.

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

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Vandermonde matrix V.

    V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

  • Wavelets Wjk(t).

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

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