 5.8.5.1.150: Showing the details of your calculations, develop:7x4  6x2
 5.8.5.1.151: Showing the details of your calculations, develop:(x + 1)2
 5.8.5.1.152: Showing the details of your calculations, develop:
 5.8.5.1.153: Showing the details of your calculations, develop:
 5.8.5.1.154: Prove that if f(x) in Example 2 is even [that is, f(x) = f(  x)], ...
 5.8.5.1.155: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.156: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.157: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.158: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.159: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.160: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.161: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.162: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.163: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.164: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.165: Find and graph (on common axes) the partial sums up to that S"'o wh...
 5.8.5.1.166: CAS EXPERIMENT. FourierBessel Series. Use Example 3 and again take...
 5.8.5.1.167: TEAM PROJECT. Orthogonality on the Entire Real Axis. Hermite Polyno...
 5.8.5.1.168: WRITING PROJECT. Orthogonality. Write a short report (23 pages) ab...
Solutions for Chapter 5.8: Orthogonal Eigenfunction Expansions
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 5.8: Orthogonal Eigenfunction Expansions
Get Full SolutionsChapter 5.8: Orthogonal Eigenfunction Expansions includes 19 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 19 problems in chapter 5.8: Orthogonal Eigenfunction Expansions have been answered, more than 46216 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their 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).

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!

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

GramSchmidt 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.

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

Lucas numbers
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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.