 4.8.1: In Exercises 18, determine which functions are solutions of the lin...
 4.8.2: In Exercises 18, determine which functions are solutions of the lin...
 4.8.3: In Exercises 18, determine which functions are solutions of the lin...
 4.8.4: In Exercises 18, determine which functions are solutions of the lin...
 4.8.5: In Exercises 18, determine which functions are solutions of the lin...
 4.8.6: In Exercises 18, determine which functions are solutions of the lin...
 4.8.7: In Exercises 18, determine which functions are solutions of the lin...
 4.8.8: In Exercises 18, determine which functions are solutions of the lin...
 4.8.9: In Exercises 916, find the Wronskian for the set of functions.
 4.8.10: In Exercises 916, find the Wronskian for the set of functions.
 4.8.11: In Exercises 916, find the Wronskian for the set of functions.
 4.8.12: In Exercises 916, find the Wronskian for the set of functions.
 4.8.13: In Exercises 916, find the Wronskian for the set of functions.
 4.8.14: In Exercises 916, find the Wronskian for the set of functions.
 4.8.15: In Exercises 916, find the Wronskian for the set of functions.
 4.8.16: In Exercises 916, find the Wronskian for the set of functions.
 4.8.17: In Exercises 1724, test the given set of solutions for linear indep...
 4.8.18: In Exercises 1724, test the given set of solutions for linear indep...
 4.8.19: In Exercises 1724, test the given set of solutions for linear indep...
 4.8.20: In Exercises 1724, test the given set of solutions for linear indep...
 4.8.21: In Exercises 1724, test the given set of solutions for linear indep...
 4.8.22: In Exercises 1724, test the given set of solutions for linear indep...
 4.8.23: In Exercises 1724, test the given set of solutions for linear indep...
 4.8.24: In Exercises 1724, test the given set of solutions for linear indep...
 4.8.25: Find the general solution of the differential equation from Exercis...
 4.8.26: Find the general solution of the differential equation from Exercis...
 4.8.27: Find the general solution of the differential equation from Exercis...
 4.8.28: Find the general solution of the differential equation from Exercis...
 4.8.29: Prove that is the general solution of
 4.8.30: Prove that the set is linearly independent if and only if
 4.8.31: Prove that the set is linearly independent.
 4.8.32: Prove that the set where is linearly independent.
 4.8.33: Writing Is the sum of two solutions of a nonhomogeneous linear diff...
 4.8.34: Writing Is the scalar multiple of a solution of a nonhomogeneous li...
 4.8.35: In Exercises 3552, identify and sketch the graph.
 4.8.36: In Exercises 3552, identify and sketch the graph.
 4.8.37: In Exercises 3552, identify and sketch the graph.
 4.8.38: In Exercises 3552, identify and sketch the graph.
 4.8.39: In Exercises 3552, identify and sketch the graph.
 4.8.40: In Exercises 3552, identify and sketch the graph.
 4.8.41: In Exercises 3552, identify and sketch the graph.
 4.8.42: In Exercises 3552, identify and sketch the graph.
 4.8.43: In Exercises 3552, identify and sketch the graph.
 4.8.44: In Exercises 3552, identify and sketch the graph.
 4.8.45: In Exercises 3552, identify and sketch the graph.
 4.8.46: In Exercises 3552, identify and sketch the graph.
 4.8.47: In Exercises 3552, identify and sketch the graph.
 4.8.48: In Exercises 3552, identify and sketch the graph.
 4.8.49: In Exercises 3552, identify and sketch the graph.
 4.8.50: In Exercises 3552, identify and sketch the graph.
 4.8.51: In Exercises 3552, identify and sketch the graph.
 4.8.52: In Exercises 3552, identify and sketch the graph.
 4.8.53: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.54: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.55: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.56: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.57: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.58: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.59: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.60: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.61: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.62: In Exercises 5362, perform a rotation of axes to eliminate the ter...
 4.8.63: In Exercises 6366, perform a rotation of axes to eliminate the ter...
 4.8.64: In Exercises 6366, perform a rotation of axes to eliminate the ter...
 4.8.65: In Exercises 6366, perform a rotation of axes to eliminate the ter...
 4.8.66: In Exercises 6366, perform a rotation of axes to eliminate the ter...
 4.8.67: Prove that a rotation of will eliminate the xyterm from the equation
 4.8.68: Prove that a rotation of where will eliminate the xyterm from the ...
 4.8.69: For the equation define the matrix as Prove that if then the graph ...
 4.8.70: For the equation in Exercise 69, define the matrix as _A_ _ 0, and ...
Solutions for Chapter 4.8: Applications of Vector Spaces
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 4.8: Applications of Vector Spaces
Get Full SolutionsChapter 4.8: Applications of Vector Spaces includes 70 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6. Since 70 problems in chapter 4.8: Applications of Vector Spaces have been answered, more than 18655 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762.

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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

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

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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 .

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 matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

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).

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 ().

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.