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# Solutions for Chapter 4.6: Systems of Linear Differential Equations

## Full solutions for Advanced Engineering Mathematics | 6th Edition

ISBN: 9781284105902

Solutions for Chapter 4.6: Systems of Linear Differential Equations

Solutions for Chapter 4.6
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##### ISBN: 9781284105902

Since 23 problems in chapter 4.6: Systems of Linear Differential Equations have been answered, more than 36877 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Chapter 4.6: Systems of Linear Differential Equations includes 23 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Column space C (A) =

space of all combinations of the columns of A.

• Complex conjugate

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

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Hankel matrix H.

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

• Indefinite matrix.

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

• 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.

• Linear combination cv + d w or L C jV j.

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Outer product uv T

= column times row = rank one matrix.

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• Projection matrix P onto subspace S.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• Solvable system Ax = b.

The right side b is in the column space of A.

• Wavelets Wjk(t).

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

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