- 31.31.1: In 31.1 and 31.2, assume that F is an ordered field, a, b E F, and ...
- 31.31.2: In 31.1 and 31.2, assume that F is an ordered field, a, b E F, and ...
- 31.31.3: Prove that if a is rational and b is irrational, then a + b is irra...
- 31.31.4: Prove that if a is rational, a =1= 0, and b is irrational, then ab ...
- 31.31.5: Prove that if u is a least upper bound for a subset 5 of JR, then 2...
- 31.31.6: Prove that if u is a least upper bound for a subset 5 of JR, then 3...
- 31.31.7: Explain why 0.9 = I.
- 31.31.8: Express 1.935 as a fraction
- 31.31.9: Prove that a decimal number that terminates represents a rational n...
- 31.31.10: Prove that a decimal number that becomes periodic represents a rati...
- 31.31.11: Determine the decimal representations of each of the following numb...
- 31.31.12: Explain why the decimal representation of a rational number must te...
- 31.31.13: Prove that if p is a prime, then .JP is irrational. ( 13.19 gives a...
- 31.31.14: (a) Write definitions of lower bound and greatest lower bound for a...
- 31.31.15: Prove the last part of Theorem 31.3, that is, prove that each compl...
- 31.31.16: Prove that if a and b are two distinct positive real numbers, then ...
- 31.31.17: Prove that a subset of an ordered field has at most one least upper...
- 31.31.18: Prove that if a and b are positive real numbers, then there exists ...
- 31.31.19: Prove that if a, b E iQ, and a > b, then there. are infinitely many...
- 31.31.20: Prove that the statement in 1 .19 is true if" <Q is replaced by any...
- 31.31.21: Prove that if a, b E JR, and a > b, then there exists a rational nu...
- 31.31.22: Assume a, bE JR and a > b.(a) Prove thata-b a> b+ viz > b.(b) Use p...
- 31.31.23: True or false: If a is irrational, then a -I is irrational.
- 31.31.24: Prove that every real number is a least upper bound of some set of ...
- 31.31.25: Give examples to show that if a and b are irrational, thenab may be...
- 31.31.26: Prove that the order on iQ given by Theorem 31.2 is the only one th...
Solutions for Chapter 31: ORDERED FIELDS. THE FIELD OF REAL NUMBERS
Full solutions for Modern Algebra: An Introduction | 6th Edition
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.
Upper triangular systems are solved in reverse order Xn to Xl.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
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).
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].
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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.
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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).
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.