×
×

# Solutions for Chapter 6.5: Impulse Functions

## Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition

ISBN: 9781119256007

Solutions for Chapter 6.5: Impulse Functions

Solutions for Chapter 6.5
4 5 0 234 Reviews
24
4
##### ISBN: 9781119256007

This expansive textbook survival guide covers the following chapters and their solutions. Since 19 problems in chapter 6.5: Impulse Functions have been answered, more than 13720 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: 9781119256007. Chapter 6.5: Impulse Functions includes 19 full step-by-step solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11.

Key Math Terms and definitions covered in this textbook
• 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.

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

• Dimension of vector space

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

• Free variable Xi.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

• Hankel matrix H.

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

• Hilbert matrix hilb(n).

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• 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 inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Subspace S of V.

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

• Wavelets Wjk(t).

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

×