 6.3.1: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.2: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.3: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.4: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.5: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.6: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.7: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.8: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.9: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.10: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.11: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.12: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.13: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.14: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.15: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.16: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.17: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.18: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.19: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.20: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.21: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.22: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.23: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.24: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.25: Suppose that F(s) = L{f(t)} exists for s > a 0.(a) Show that if c i...
 6.3.26: In each of 26 through 29, use the results of to find the inverse La...
 6.3.27: In each of 26 through 29, use the results of to find the inverse La...
 6.3.28: In each of 26 through 29, use the results of to find the inverse La...
 6.3.29: In each of 26 through 29, use the results of to find the inverse La...
 6.3.30: In each of 30 through 33, find the Laplace transform of the given f...
 6.3.31: In each of 30 through 33, find the Laplace transform of the given f...
 6.3.32: In each of 30 through 33, find the Laplace transform of the given f...
 6.3.33: In each of 30 through 33, find the Laplace transform of the given f...
 6.3.34: Let f satisfy f(t + T) = f(t) for all t 0 and for some fixed positi...
 6.3.35: In each of 35 through 38, use the result of to find the Laplace tra...
 6.3.36: In each of 35 through 38, use the result of to find the Laplace tra...
 6.3.37: In each of 35 through 38, use the result of to find the Laplace tra...
 6.3.38: In each of 35 through 38, use the result of to find the Laplace tra...
 6.3.39: (a) If f(t) = 1 u1(t), find L{f(t)}; compare with 30. Sketch the gr...
 6.3.40: Consider the function p defined byp(t) =t, 0 t < 1,2 t, 1 t < 2; p(...
Solutions for Chapter 6.3: Step Functions
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 6.3: Step Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.3: Step Functions includes 40 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Since 40 problems in chapter 6.3: Step Functions have been answered, more than 12541 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).