 1.1.1: Let S = {w, x, y, z} and T = {I, 2, 3, 4}, and define a: S > T and...
 1.1.2: Leta, f3, and y be mappings from Z to Z defined by a(n) = 2n, f3(n)...
 1.1.3: For 1.31.6, assume S = {x, y, z} and T = {I, 2, 3}. From Example 1...
 1.1.4: For 1.31.6, assume S = {x, y, z} and T = {I, 2, 3}. From Example 1...
 1.1.5: For 1.31.6, assume S = {x, y, z} and T = {I, 2, 3}. From Example 1...
 1.1.6: For 1.31.6, assume S = {x, y, z} and T = {I, 2, 3}. From Example 1...
 1.1.7: For 1.71.1 0, assume that Sand T are sets, a : S > T, and f3 : S ...
 1.1.8: For 1.71.1 0, assume that Sand T are sets, a : S > T, and f3 : S ...
 1.1.9: For 1.71.1 0, assume that Sand T are sets, a : S > T, and f3 : S ...
 1.1.10: For 1.71.1 0, assume that Sand T are sets, a : S > T, and f3 : S ...
 1.1.11: Each f in 1.111.16 defines a mapping from lR (or a subset of lR) t...
 1.1.12: Each f in 1.111.16 defines a mapping from lR (or a subset of lR) t...
 1.1.13: Each f in 1.111.16 defines a mapping from lR (or a subset of lR) t...
 1.1.14: Each f in 1.111.16 defines a mapping from lR (or a subset of lR) t...
 1.1.15: Each f in 1.111.16 defines a mapping from lR (or a subset of lR) t...
 1.1.16: Each f in 1.111.16 defines a mapping from lR (or a subset of lR) t...
 1.1.17: In 1.17 and 1.18 let A denote the set of odd natural numbers, B the...
 1.1.18: In 1.17 and 1.18 let A denote the set of odd natural numbers, B the...
 1.1.19: In 1.19 and 1.20,for each n E Z the mapping f" : Z > Z is defined ...
 1.1.20: In 1.19 and 1.20,for each n E Z the mapping f" : Z > Z is defined ...
 1.1.21: Assume that Sand T are finite sets containing m and n elements, res...
 1.1.22: (a) How many mappings are there from a twoelement set onto a twoe...
 1.1.23: A mapping f : IR + IR is onto iff each horizontal line (line paral...
 1.1.24: For each ordered pair (a, b) of integers define a mapping aa.b : Z ...
 1.1.25: With f3 as defined in Example 1.10, for each n EN the equation f3(x...
 1.1.26: Prove that there is a mapping from a set to itself that is onetoo...
 1.1.27: Prove that if a : S + T and A and B are subsets of S, then a(A U B...
 1.1.28: (a) Prove that if a : S + T, and A and B are subsets of S, then a(...
 1.1.29: (a) Prove that if a : S + T, and A and B are subsets of S, then a(...
 1.1.30: Using the definition of infinite from Example 1.10, prove that if a...
 1.1.31: Define a onetoone mapping from the set of natural numbers onto th...
Solutions for Chapter 1: MAPPINGS
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 1: MAPPINGS
Get Full SolutionsChapter 1: MAPPINGS includes 31 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. Since 31 problems in chapter 1: MAPPINGS have been answered, more than 8826 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

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].

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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].

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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(DÂ» O.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.