 8.8.1: Draw figures like those in Figure 8.4 to verify the entries for /LI...
 8.8.2: Draw figures like those in Figure 8.4 to verify the entries for /L2...
 8.8.3: Determine the group of symmetries of an equilateral triangle.
 8.8.4: Determine the group of symmetries of an isosceles triangle.
 8.8.5: Determine the group of symmetries of a regular pentagon. (It will h...
 8.8.6: Determine the permutation of the vertices of the square abed corres...
 8.8.7: The permutation (ab)(c)(d) of the vertices of the square abed (Figu...
 8.8.8: Consider the mapping T t+ M(T) from the set of subsets of a plane ...
 8.8.9: Using the notation of Example 8.1, determine the group of symmetrie...
 8.8.10: Consider symmetry under the motions in Example 8.1. As geometric ob...
 8.8.11: Determine the group of symmetries of each of the following figures....
 8.8.12: Determine the group of symmetries of each of the following figures....
 8.8.13: Determine the group of symmetries of each of the following figures....
 8.8.14: Determine the group of symmetries of each of the following figures....
 8.8.15: Determine the group of symmetries of each of the following figures....
 8.8.16: Determine the group of symmetries of each of the following figures....
 8.8.17: [Refer to Figure 57.2 for this problem. It is part of Example 57.3,...
 8.8.18: [For this problem, refer to the parenthetical statement at the end ...
Solutions for Chapter 8: GROUPS AND SYMMETRY
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 8: GROUPS AND SYMMETRY
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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.)

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).