- 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: Vector Spaces
Full solutions for Contemporary Abstract Algebra | 8th Edition
ISBN: 9781133599708
Solutions for Chapter 19
22
3
Summary of Chapter 19: Vector Spaces
In this chapter, we provide a concise review of this material.
This expansive textbook survival guide covers the following chapters and their solutions. Chapter 19: Vector Spaces includes 32 full step-by-step solutions. Since 32 problems in chapter 19: Vector Spaces have been answered, more than 229260 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.
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Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.
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Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
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Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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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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Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
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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.
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Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
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Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
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Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.