Algebra and Trigonometry: Real Mathematics, Real People 7th Edition - Solutions by Chapter

Parentheses can be removed to leave ABC.

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.)

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

Remove row i and column j; multiply the determinant by (-I)i + j •

If diagonalizable, they share n eigenvectors.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

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.

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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

P = aaT laTa has rank l.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

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

The right side b is in the column space of A.

Spectral radius = max of IAi I.

Columns of n by n identity matrix (written i ,j ,k in R3).

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.