 6.4.1: Define SQRT: by SQRT a) Is SQRT: operation preserving?(b) Is SQRT: ...
 6.4.2: Define SQR: by SQR(a) Is SQR: operation preserving?(b) Is SQR: oper...
 6.4.3: Define on by setting(a) Show that is an algebraic system.(b) Show t...
 6.4.4: Let be the set of all realvalued integrable functions defined on t...
 6.4.5: Let f: and be OP maps.(a) Prove that is an OP map.(b) Prove that if...
 6.4.6: Let be the set of all matrices with real entries. Define Det: byDet...
 6.4.7: Let Conj: be the conjugate mapping for complex numbers given byConj...
 6.4.8: Let f be a function from set A to set B. Let f and be the induced f...
 6.4.9: Prove Theorem 6.4.3.
 6.4.10: (a) Show that any two groups of order 2 are isomorphic.(b) Show tha...
 6.4.11: Let and be the sets of integer multiples of 3 and 6, respectively. ...
 6.4.12: Let and be the groups in Exercise 12 and let g be the functionfrom ...
 6.4.13: Let be the group with the operation table shown here.a b ca a b cb ...
 6.4.14: Let and (a) Prove that the function given by is welldefined and is ...
 6.4.15: Let and Defineby f (xq) = [4x]. (a) Prove that f is a welldefined ...
 6.4.16: Let and be groups, i be the identity element for H, andbe a homomor...
 6.4.17: Show that and are isomorphic.
 6.4.18: Is isomorphic to Explain.
 6.4.19: Prove that the relation of isomorphism is an equivalence relation. ...
 6.4.20: Use the method of proof of Cayleys Theorem to find a group of permu...
 6.4.21: Assign a grade of A (correct), C (partially correct), or F (failure...
 6.4.22: Claim. Let and be OP maps.Then the composite is an OP map.Proof.
Solutions for Chapter 6.4: Operation Preserving Maps
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
Solutions for Chapter 6.4: Operation Preserving Maps
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.4: Operation Preserving Maps includes 22 full stepbystep solutions. A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780495562023. Since 22 problems in chapter 6.4: Operation Preserving Maps have been answered, more than 5555 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: A Transition to Advanced Mathematics, edition: 7.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of 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.

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

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

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.

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.