 7.3.1: In each of 1 through 5, either solve the given system of equations,...
 7.3.2: In each of 1 through 5, either solve the given system of equations,...
 7.3.3: In each of 1 through 5, either solve the given system of equations,...
 7.3.4: In each of 1 through 5, either solve the given system of equations,...
 7.3.5: In each of 1 through 5, either solve the given system of equations,...
 7.3.6: In each of 6 through 9, determine whether the members of the given ...
 7.3.7: In each of 6 through 9, determine whether the members of the given ...
 7.3.8: In each of 6 through 9, determine whether the members of the given ...
 7.3.9: In each of 6 through 9, determine whether the members of the given ...
 7.3.10: Suppose that each of the vectors x(1) , . . . , x(m) has n componen...
 7.3.11: x(1) (t) = (et, 2et ), x(2) (t) = (et , et ), x(3) (t) = (3et, 0) 1
 7.3.12: x(1) (t) = (2 sin t, sin t), x(2) (t) = (sin t, 2 sin t) 1
 7.3.13: Let x(1) (t) = _et tet _, x(2) (t) = _1t _ . Show that x(1) (t) and...
 7.3.14: In each of 14 through 20, find all eigenvalues and eigenvectors of ...
 7.3.15: In each of 14 through 20, find all eigenvalues and eigenvectors of ...
 7.3.16: In each of 14 through 20, find all eigenvalues and eigenvectors of ...
 7.3.17: In each of 14 through 20, find all eigenvalues and eigenvectors of ...
 7.3.18: In each of 14 through 20, find all eigenvalues and eigenvectors of ...
 7.3.19: In each of 14 through 20, find all eigenvalues and eigenvectors of ...
 7.3.20: In each of 14 through 20, find all eigenvalues and eigenvectors of ...
 7.3.21: a. Suppose that A is a realvalued n n matrix. Show that (Ax, y) = ...
 7.3.22: Suppose that, for a given matrix A, there is a nonzero vector x suc...
 7.3.23: Suppose that detA = 0 and that Ax = b has solutions. Show that (b, ...
 7.3.24: Suppose that detA = 0 and that x = x(0) is a solution of Ax = b. Sh...
 7.3.25: Suppose that detA = 0 and that y is a solution of Ay = 0. Show that...
 7.3.26: Prove that = 0 is an eigenvalue of A if and only if A is singular. 2
 7.3.27: In this problem we show that the eigenvalues of a Hermitian matrix ...
 7.3.28: Show that if 1 and 2 are eigenvalues of a Hermitian matrix A, and i...
 7.3.29: Show that if 1 and 2 are eigenvalues of any matrix A, and if 1 _= 2...
Solutions for Chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors includes 29 full stepbystep solutions. Since 29 problems in chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors have been answered, more than 13346 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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