 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 thatab 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
ISBN: 9780470384435
Solutions for Chapter 31: ORDERED FIELDS. THE FIELD OF REAL NUMBERS
Get Full SolutionsSince 26 problems in chapter 31: ORDERED FIELDS. THE FIELD OF REAL NUMBERS have been answered, more than 8295 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Chapter 31: ORDERED FIELDS. THE FIELD OF REAL NUMBERS includes 26 full stepbystep solutions.

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.

Back substitution.
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. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + 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.

Graph G.
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, ... , AjIb. 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)ยท(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pascal matrix
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+ (MoorePenrose 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. AI 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 AI are BT AT and (AT)I.