- 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 finite-dimensional 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
22
3
This expansive textbook survival guide covers the following chapters and their solutions. Chapter 19 includes 32 full step-by-step solutions. Since 32 problems in chapter 19 have been answered, more than 51859 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708.
Key Math Terms and definitions covered in this textbook
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
-
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.
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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).
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Column space C (A) =
space of all combinations of the columns of A.
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Companion matrix.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
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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.
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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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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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).
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Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.
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Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
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Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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Solvable system Ax = b.
The right side b is in the column space of A.