- 8.5.1: Integrate the trig identity 2 cos j x cos kx = cos(j + k)x + cos(j ...
- 8.5.2: Show that 1, x, and x 2 - 1 are orthogonal, when the integration is...
- 8.5.3: Find a vector (WI, W2, W3, . .. ) that is orthogonal to v = (I,!, i...
- 8.5.4: The first three Legendre polynomials are 1, x, and x 2 -1. Choose e...
- 8.5.5: For the square wave I(x) in Example 3, show that 2n ' fo I(x) cosx ...
- 8.5.6: The square wave has 11/112 = 2](. Then (6) gives what remarkable su...
- 8.5.7: Graph the square wave. Then graph by hand the sum of two sine terms...
- 8.5.8: Find the lengths of these vectors in Hilbert space: (a) v = (JI, ~,...
- 8.5.9: Compute the Fourier coefficients ak and bk for f(x) defined from 0 ...
- 8.5.10: When f(x) has period 2n, why is its integral from -n to n the same ...
- 8.5.11: From trig identities find the only two terms in the Fourier series ...
- 8.5.12: The functions 1, cos x, sin x, cos 2x, sin 2x, ... are a basis for ...
- 8.5.13: Find the Fourier coefficients ak and bk of the square pulse F(x) ce...
Solutions for Chapter 8.5: Fourier Series: Linear Algebra for Functions
Full solutions for Introduction to Linear Algebra | 4th Edition
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
Invert A by row operations on [A I] to reach [I A-I].
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Outer product uv T
= column times row = rank one matrix.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.