- 1.7.1: (a) Prove that if n is an integer and is odd, then is divisibleby 4...
- 1.7.2: Prove that(a) for all integers n, is even.(b) for all odd integers ...
- 1.7.4: (a) Prove that if x is rational and y is irrational, then is irrati...
- 1.7.5: (a) Prove that except for two points on the circle, if is on the ci...
- 1.7.7: Prove that for all real numbers x,(a) if then(b) if or then (x 1)(x...
- 1.7.8: Prove or disprove: (a) Every point inside the circle is inside the ...
- 1.7.10: (a) Let a and b be integers and Prove that if then when b isdivided...
- 1.7.11: For each pair of integers, list all positive and negative common di...
- 1.7.13: Find and integers x and y such that
- 1.7.14: Let a, b, and c be natural numbers and Prove that (a) if c divides ...
- 1.7.16: Prove that for every prime p and for all natural numbers a, (a) p d...
- 1.7.17: Let q be a natural number greater than 1 with the property that q d...
- 1.7.18: Let a and b be nonzero integers that are relatively prime, and let ...
- 1.7.19: . Let a and b be nonzero integers and Let and Showthat if and is a ...
- 1.7.20: For nonzero integers a and b, the integer n is a common multiple of...
- 1.7.21: . Let a, b, and c be natural numbers, and Provethat (a) a divides b...
- 1.7.22: Let a and b be integers, and let Use the Division Algorithm toprove...
- 1.7.23: Assign a grade of A (correct), C (partially correct), or F (failure...
Solutions for Chapter 1.7: Additional Examples of Proofs
Full solutions for A Transition to Advanced Mathematics | 7th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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].
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.
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.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).