 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  10th Edition
ISBN: 9780470458327
Solutions for Chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors includes 34 full stepbystep solutions. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. This expansive textbook survival guide covers the following chapters and their solutions. Since 34 problems in chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors have been answered, more than 11384 students have viewed full stepbystep solutions from this chapter.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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].

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.