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Fundamentals of Differential Equations and Boundary Value Problems 6th Edition - Solutions by Chapter
Full solutions for Fundamentals of Differential Equations and Boundary Value Problems | 6th Edition
ISBN: 9780321747747
Fundamentals of Differential Equations and Boundary Value Problems | 6th Edition - Solutions by Chapter
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This textbook survival guide was created for the textbook: Fundamentals of Differential Equations and Boundary Value Problems, edition: 6. Since problems from 10 chapters in Fundamentals of Differential Equations and Boundary Value Problems have been answered, more than 2459 students have viewed full step-by-step answer. The full step-by-step solution to problem in Fundamentals of Differential Equations and Boundary Value Problems were answered by , our top Math solution expert on 11/14/17, 08:38PM. Fundamentals of Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780321747747. This expansive textbook survival guide covers the following chapters: 10.
Key Math Terms and definitions covered in this textbook
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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!
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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.)
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Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.
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Cofactor Cij.
Remove row i and column j; multiply the determinant by (-I)i + j •
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Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
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Dimension of vector space
dim(V) = number of vectors in any basis for V.
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Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
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Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
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Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
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Toeplitz matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.