 40.40.1: Prove that if F is a field with unity e, and I is an ideal of F[x],...
 40.40.2: Prove that if F is a field and p(x), f(x) E F[x], with p(x) irreduc...
 40.40.3: Verify that the mapping b H I + b in the proof of Theorem 40.2 is ...
 40.40.4: Prove that the mapping e in Example .40.1 is onetoone and onto, a...
 40.40.5: Prove that if F is a subfield of a field E, and c E E, then e : F[x...
 40.40.6: Ife is any homomorphism of the type in 40.5, then Kere = (b(x))for ...
 40.40.7: Ife is any homomorphism of the type in 40.5, then Kere = (b(x))for ...
 40.40.8: Ife is any homomorphism of the type in 40.5, then Kere = (b(x))for ...
 40.40.9: Ife is any homomorphism of the type in 40.5, then Kere = (b(x))for ...
 40.40.10: Ife is any homomorphism of the type in 40.5, then Kere = (b(x))for ...
 40.40.11: Suppose thatF is a field and f(x), g(x) E F[x]. Prove that (f(x)) =...
 40.40.12: Prove that if F is a field and I is an ideal of F[x], then there is...
 40.40.13: (a) Prove or disprove that if (f(x)) = (g(x, then deg f(x) = deg g(...
 40.40.14: True or false: If f(x) E (g(x)) and deg f(x) = deg g(x), then (f(x ...
 40.40.15: The proof of Theorem 40.3 uses the Least Integer Principle. Where?
 40.40.16: Determine all of the prime ideals of F[x], where F is a field. (See...
Solutions for Chapter 40: QUOTIENT RINGS OF F[X]
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 40: QUOTIENT RINGS OF F[X]
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.