- Chapter 0:
- Chapter 1:
- Chapter 10:
- Chapter 11:
- Chapter 12:
- Chapter 13:
- Chapter 14:
- Chapter 15:
- Chapter 16:
- Chapter 17:
- Chapter 18:
- Chapter 19:
- Chapter 2:
- Chapter 20:
- Chapter 21:
- Chapter 22:
- Chapter 23:
- Chapter 24:
- Chapter 25:
- Chapter 26:
- Chapter 27:
- Chapter 28:
- Chapter 29:
- Chapter 3:
- Chapter 30:
- Chapter 31:
- Chapter 32:
- Chapter 33:
- Chapter 4:
- Chapter 5:
- Chapter 6:
- Chapter 7:
- Chapter 8:
- Chapter 9:
Contemporary Abstract Algebra 8th Edition - Solutions by Chapter
Full solutions for Contemporary Abstract Algebra | 8th Edition
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.)
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
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
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Gram-Schmidt 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.
A sequence of steps intended to approach the desired solution.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
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.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
A directed graph that has constants Cl, ... , Cm associated with the edges.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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