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# Solutions for Chapter 10.2: Fourier Series

## Full solutions for Elementary Differential Equations and Boundary Value Problems | 9th Edition

ISBN: 9780470383346

Solutions for Chapter 10.2: Fourier Series

Solutions for Chapter 10.2
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##### ISBN: 9780470383346

This expansive textbook survival guide covers the following chapters and their solutions. Since 29 problems in chapter 10.2: Fourier Series have been answered, more than 12844 students have viewed full step-by-step solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Chapter 10.2: Fourier Series includes 29 full step-by-step solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9.

Key Math Terms and definitions covered in this textbook
• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Change of basis matrix M.

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.)

• Cofactor Cij.

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

• Complex conjugate

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

• Diagonal matrix D.

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

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Independent vectors VI, .. " vk.

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.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• 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.

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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