- 15.15.1: Verify each of the following equalities for subgroups of 2': (-20,8...
- 15.15.2: Verify each of the following equalities for subgroups of 2':(24, -3...
- 15.15.3: Verify each of the following equalities for subgroups of S6. ((1 4 ...
- 15.15.4: Verify each of the following equalities for subgroups of S6. (1 2 3...
- 15.15.5: Determine the elements in each of the following subgroups of the gr...
- 15.15.6: Determine the elements in each of the following subgroups of the gr...
- 15.15.7: Determine the elements in each of the following subgroups of the gr...
- 15.15.8: Determine the elements in each of the following subgroups of the gr...
- 15.15.9: Simplify the following expression in 2':4 x S4.(, (1 2 3-1(, ...
- 15.15.10: Simplify the following expression in A x B, where A is the group in...
- 15.15.11: Construct a Cayley table for the group in Example 15.4. Show that t...
- 15.15.12: Construct a Cayley table for 2':2 X 2':3. Show that the group is cy...
- 15.15.13: The subgroup ((1 4 3 2), (2 4 of S4 has order 8. Determine its elem...
- 15.15.14: f 0 denotes the empty set, what is (0) (in any group G)?
- 15.15.15: Find necessary and sufficient conditions on a subset S of a group G...
- 15.15.16: Prove the associative law for the direct prOduct A x B of groups A ...
- 15.15.17: Prove that A x (e) is a subgroup of Ax B.
- 15.15.18: Prove that A x B is Abelian iff both A and B are Abelian.
- 15.15.19: Give an example to show that a direct product of two cyclic groups ...
- 15.15.20: Prove that if A is a subgroup of G and B is a subgroup of H, then A...
- 15.15.21: (a) List the elements of S3 x 2':2. (b) List the elements of the cy...
- 15.15.22: (a) List the elements in the subgroup ((, )) of 2:4 x 2:s. (T...
- 15.15.23: Prove that if a, b E 2:, then (a, b) = (d), where d is the greatest...
- 15.15.24: Prove that if a, b E 2:, then (a) n (b) = (m), where m is the least...
- 15.15.25: Prove that ([al) = 2:n iff (a, n) = 1, where (a, n) denotes the gre...
- 15.15.26: Prove that if (a, n) = d, then ([al) = ([d]) in 2:n. [Here (a, n) d...
- 15.15.27: Prove that ([aJ) = ([b]) in 2:" iff (a, n) = (b, n). (See 15.26.)
- 15.15.28: Prove that if A is a group, then ((a, a) : a E A} is a subgroup of ...
- 15.15.29: Each subgroup of 2: (operation +) is cyclic. Prove this by assuming...
- 15.15.30: Prove that if G is a group with operation *, and S is a nonempty su...
Solutions for Chapter 15: GENERATORS. DIRECT PRODUCTS
Full solutions for Modern Algebra: An Introduction | 6th Edition
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.
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.)
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
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.
Every v in V is orthogonal to every w in W.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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!
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).