 3.1.1: Let T be the relation Find
 3.1.2: Find the domain and range for the relation W on given by
 3.1.3: Sketch the graph of each relation in Exercise 2.
 3.1.4: The inverse of may be expressed in theform the set of all pairs sub...
 3.1.5: . Let andFind
 3.1.6: Find these composites for the relations defined in Exercise 4.
 3.1.7: Give the digraphs for these relations on the set {1, 2, 3}.
 3.1.8: Let Give an example of relations R, S, and T on A such that
 3.1.9: 9. Let R be a relation from A to B and S be a relation from B to C....
 3.1.10: Complete the proof of Theorem 3.1.3.
 3.1.11: Show by example that may be false.
 3.1.12: different relations from A to B.
 3.1.13: . (a) Let R be a relation from A to B. For define the vertical sect...
 3.1.14: We may define ordered triples in terms of ordered pairs by saying t...
 3.1.15: Assign a grade of A (correct), C (partially correct), or F (failure...
Solutions for Chapter 3.1: Relations and Partitions
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
Solutions for Chapter 3.1: Relations and Partitions
Get Full SolutionsA Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780495562023. Since 15 problems in chapter 3.1: Relations and Partitions have been answered, more than 5793 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: A Transition to Advanced Mathematics, edition: 7. Chapter 3.1: Relations and Partitions includes 15 full stepbystep solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.