 7.1.1: Find four upper bounds (if any exist) for each of the following sets.
 7.1.2: Find a lower bound in (if one exists) for each of the sets in Exerc...
 7.1.3: Find the supremum and infimum, if they exist, of each of the follow...
 7.1.4: Let A and B be subsets of Prove that (a) if A is bounded above and ...
 7.1.5: Let x be an upper bound for Prove that(a) if then y is an upper bou...
 7.1.6: Let Prove that (a) if A is bounded above, then is not bounded above...
 7.1.7: Give an example of a set for which both A and are unbounded abovean...
 7.1.8: Let Prove that (a) if exists, then it is unique. That is, if x and ...
 7.1.9: Let Prove that(a) if and both exist, then(b) if and both exist, the...
 7.1.10: Formulate and prove a characterization of greatest lower bounds sim...
 7.1.11: If possible, give an example of(a) a set such that and(b) a set A s...
 7.1.12: Give an example of a set of rational numbers that has a rational lo...
 7.1.13: Let Prove that (a) if exists, then sup (A) = inf {u: u is an upper ...
 7.1.14: Let A and B be subsets of (a) Prove that if and exist, then exists ...
 7.1.15: (a) Give an example of sets A and B of real numbers such thatand(b)...
 7.1.16: (a) Give an example of sets A and B of real numbers such thatand(b)...
 7.1.17: Use the completeness property of to prove the Archimedean Principle...
 7.1.18: This exercise shows that every irrational number is missing from Le...
 7.1.19: Prove that an ordered field F is complete every nonempty subset of ...
 7.1.20: Let F be an ordered field and Prove that(a) exactly one of or is tr...
 7.1.21: Assign a grade of A (correct), C (partially correct), or F (failure...
Solutions for Chapter 7.1: Completeness of the Real Numbers
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
Solutions for Chapter 7.1: Completeness of the Real Numbers
Get Full SolutionsThis textbook survival guide was created for the textbook: A Transition to Advanced Mathematics, edition: 7. Chapter 7.1: Completeness of the Real Numbers includes 21 full stepbystep solutions. A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780495562023. This expansive textbook survival guide covers the following chapters and their solutions. Since 21 problems in chapter 7.1: Completeness of the Real Numbers have been answered, more than 5818 students have viewed full stepbystep solutions from this chapter.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Solvable system Ax = b.
The right side b is in the column space of A.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).