 19.1E: Verify that each of the sets in Examples 1– 4 satisfies the axioms ...
 19.2E: (Subspace Test) Prove that a nonempty subset U of a vector space V ...
 19.3E: Verify that the set in Example 6 is a subspace. Find a basis for th...
 19.4E: Verify that the set defined in Example 7 is a subspace.Reference:
 19.5E: Determine whether or not the set {(2, –1, 0), (1, 2, 5), (7, –1, 5)...
 19.6E: Determine whether or not the set is linearly independent over Z5.
 19.7E: If {u, v, w} is a linearly independent subset of a vector space, sh...
 19.8E: If {v1, v2, . . . , vn} is a linearly dependent set of vectors, pro...
 19.9E: (Every spanning collection contains a basis.) If {v1, v2, . . . , v...
 19.10E: (Every independent set is contained in a basis.) Let V be a finite ...
 19.11E: V is a vector space over F of dimension 5 and U and W are subspaces...
 19.12E: Show that the solution set to a system of equations of the form whe...
 19.13E: Let V be the set of all polynomials over Q of degree 2 together wit...
 19.14E: Let V = R3 and W = {(a, b, c) ? V  a2 + b2 = c2}. Is W a subspace ...
 19.15E: Let V = R3 and W = {(a, b, c) ? V  a + b = c}. Is W a subspace of ...
 19.16E: Let . Prove that V is a vector space over Q, and find a basis for V...
 19.17E: Verify that the set V in Example 9 is a vector space over R.REFERNCE:
 19.18E: Let P = {(a, b, c)  a, b, c [ R, a = 2b + 3c}. Prove that P is a s...
 19.19E: Let B be a subset of a vector space V. Show that B is a basis for V...
 19.20E: If U is a proper subspace of a finitedimensional vector space V, s...
 19.21E: Referring to the proof of Theorem 19.1, prove that {w1, u2, . . . ,...
 19.22E: If V is a vector space of dimension n over the field Zp, how many e...
 19.23E: Let S = {(a, b, c, d)  a, b, c, d ? R, a = c, d = a+ b}. Find a ba...
 19.24E: Let U and W be subspaces of a vector space V. Show that U ? W is a ...
 19.25E: If a vector space has one basis that contains infinitely many eleme...
 19.26E: Let u = (2, 3, 1), v = (1, 3, 0), and w = (2, –3, 3). Since (1/2)u ...
 19.27E: Define the vector space analog of group homomorphism and ring homom...
 19.28E: Let T be a linear transformation from V to W. Prove that the image ...
 19.29E: Let T be a linear transformation of a vector space V. Prove that {v...
 19.30E: Let T be a linear transformation of V onto W. If {v1, v2, . . . , v...
 19.31E: If V is a vector space over F of dimension n, prove that V is isomo...
 19.32E: Let V be a vector space over an infinite field. Prove that V is not...
Solutions for Chapter 19: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 19
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 19 includes 32 full stepbystep solutions. Since 32 problems in chapter 19 have been answered, more than 17772 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8th. Contemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708.

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.)

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

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.

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.

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].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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

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.

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