 7.3.1: In each of 1 through 6, either solve the given system of equations,...
 7.3.2: In each of 1 through 6, either solve the given system of equations,...
 7.3.3: In each of 1 through 6, either solve the given system of equations,...
 7.3.4: In each of 1 through 6, either solve the given system of equations,...
 7.3.5: In each of 1 through 6, either solve the given system of equations,...
 7.3.6: In each of 1 through 6, either solve the given system of equations,...
 7.3.7: In each of 7 through 11, determine whether the members of the given...
 7.3.8: In each of 7 through 11, determine whether the members of the given...
 7.3.9: In each of 7 through 11, determine whether the members of the given...
 7.3.10: In each of 7 through 11, determine whether the members of the given...
 7.3.11: In each of 7 through 11, determine whether the members of the given...
 7.3.12: Suppose that each of the vectors x(1) , ... , x(m) has n components...
 7.3.13: In each of 13 and 14, determine whether the members of the given se...
 7.3.14: In each of 13 and 14, determine whether the members of the given se...
 7.3.15: Letx(1)(t) =ettet, x(2)(t) =1t.Show that x(1)(t) and x(2)(t) are li...
 7.3.16: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.17: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.18: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.19: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.20: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.21: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.22: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.23: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.24: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.25: In each of 16 through 25, find all eigenvalues and eigenvectors of ...
 7.3.26: 26 through 30 deal with the problem of solving Ax = b when detA = 0...
 7.3.27: 26 through 30 deal with the problem of solving Ax = b when detA = 0...
 7.3.28: 26 through 30 deal with the problem of solving Ax = b when detA = 0...
 7.3.29: 26 through 30 deal with the problem of solving Ax = b when detA = 0...
 7.3.30: 26 through 30 deal with the problem of solving Ax = b when detA = 0...
 7.3.31: Prove that = 0 is an eigenvalue of A if and only if A is singular.
 7.3.32: In this problem we show that the eigenvalues of a Hermitian matrix ...
 7.3.33: Show that if 1 and 2 are eigenvalues of a Hermitian matrix A, and i...
 7.3.34: 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  10th Edition
ISBN: 9780470458310
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: 9780470458310. Since 34 problems in chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors have been answered, more than 16439 students have viewed full stepbystep solutions from this chapter. Chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors includes 34 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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].

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.