 5.7.1: Use the recursion formulas to calculate (a) T4, T5 and (b) H4, H5.
 5.7.2: Let p0(x), p1(x), and p2(x) be orthogonal with respect to the inner...
 5.7.3: Show that the Chebyshev polynomials have the following properties: ...
 5.7.4: Find the best quadratic least squares approximation to ex on [1, 1]...
 5.7.5: Let p0, p1, . . . be a sequence of orthogonal polynomials and let a...
 5.7.6: Let Tn(x) denote the Chebyshev polynomial of degree n, and define U...
 5.7.7: Let Un1(x) be defined as in Exercise 6 for n 1, and define U1(x) = ...
 5.7.8: Show that the Ui s defined in Exercise 6 are orthogonal with respec...
 5.7.9: Verify that the Legendre polynomial Pn(x) satisfies the secondorde...
 5.7.10: Prove each of the following: (a) H_ n(x) = 2nHn1(x), n = 0, 1, . . ...
 5.7.11: Given a function f (x) that passes through the points (1, 2), (2,1)...
 5.7.12: Show that if f (x) is a polynomial of degree less than n, then f (x...
 5.7.13: Use the zeros of the Legendre polynomial P2(x) to obtain a twopoin...
 5.7.14: (a) For what degree polynomials will the quadrature formula in Exer...
 5.7.15: Let x1, x2, . . . , xn be distinct points in the interval [1, 1] an...
 5.7.16: Let x1, x2, . . . , xn be the roots of the Legendre polynomial Pn. ...
 5.7.17: Let Q0(x), Q1(x), . . . be an orthonormal sequence of polynomials; ...
Solutions for Chapter 5.7: Orthogonal Polynomials
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 5.7: Orthogonal Polynomials
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. Since 17 problems in chapter 5.7: Orthogonal Polynomials have been answered, more than 6801 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.7: Orthogonal Polynomials includes 17 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.