- 3.1: Between which two whole numbers is 95 ? A 10 and 11 B 9 and 10 C 8 ...
- 3.2: Keiko goes to the South Carolina State fair in Columbia, which has ...
- 3.3: Which subset of real numbers contains 3 ? A irrational numbers B ra...
- 3.4: Margo plots part of her hometown on a coordinate grid so that each ...
- 3.5: In a number game, Long was supposed to find the square root of a nu...
- 3.6: For which triangle is the relationship a 2 + b 2 = c 2 true? A a c ...
- 3.7: Between which two whole numbers is 117 ? A 8 and 9 B 9 and 10 C 10 ...
- 3.8: What is the length of the hypotenuse in the right triangle below? 3...
- 3.9: Which symbol makes the number sentence true when placed in the blan...
Solutions for Chapter 3: Math Connects: Concepts, Skills, and Problem Solving Course 3 0th Edition
Full solutions for Math Connects: Concepts, Skills, and Problem Solving Course 3 | 0th Edition
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!
A = CTC = (L.J]))(L.J]))T for positive definite A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Invert A by row operations on [A I] to reach [I A-I].
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
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 matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Every v in V is orthogonal to every w in W.
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.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.