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Solutions for Chapter 8.5: Fourier Series: Linear Algebra for Functions

Full solutions for Introduction to Linear Algebra | 4th Edition

ISBN: 9780980232714

Solutions for Chapter 8.5: Fourier Series: Linear Algebra for Functions

Chapter 8.5: Fourier Series: Linear Algebra for Functions includes 13 full step-by-step solutions. Since 13 problems in chapter 8.5: Fourier Series: Linear Algebra for Functions have been answered, more than 8145 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Diagonalization

    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.

  • Elimination.

    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

  • Fibonacci numbers

    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.

  • Gauss-Jordan method.

    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.

  • Indefinite 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)ยท(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 .

  • Normal matrix.

    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.

  • Pascal 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}.

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