 4.3.1E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.2E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.4E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.5E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.6E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.7E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.8E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.9E: Find bases for the null spaces of the matrices given in Exercises 9...
 4.3.10E: Find bases for the null spaces of the matrices given in Exercises 9...
 4.3.11E: Find a basis for the set of vectors in R3 in the plane [Hint: Think...
 4.3.12E: Find a basis for the set of vectors in on the line y = –3x.
 4.3.13E: In Exercises 13 and 14, assume that A is row equivalent to B. Find ...
 4.3.14E: In Exercises 13 and 14, assume that A is row equivalent to B. Find ...
 4.3.15E: In Exercises 15–18, find a basis for the space spanned by the given...
 4.3.16E: In Exercises 15–18, find a basis for the space spanned by the given...
 4.3.17E: In Exercises 15–18, find a basis for the space spanned by the given...
 4.3.18E: In Exercises 15–18, find a basis for the space spanned by the given...
 4.3.19E: and also let It can be verified that Use this information to find a...
 4.3.20E: It can be verified that Use this information to find a basis for
 4.3.21E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 4.3.22E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 4.3.23E: Suppose R4 = Span {v1, . . . , v4}. Explain why {v1, . . . , v4}Is ...
 4.3.24E: Let {v1, . . . ,vn}be a linearly independent set in Rn. Explain why...
 4.3.25E: and let H be the set of vectors in whose second and third entries a...
 4.3.26E: In the vector space of all realvalued functions, find a basis for ...
 4.3.27E: Let V be the vector space of functions that describe the vibration ...
 4.3.28E: (RLC circuit) The circuit in the figure consists of a resistor (R o...
 4.3.29E: Exercises 29 and 30 show that every basis for must contain exactly ...
 4.3.30E: Exercises 29 and 30 show that every basis for must contain exactly ...
 4.3.31E: Show that if is linearly dependent in V, then the set of images, , ...
 4.3.32E: Suppose that T is a onetoone transformation, so that an equation ...
 4.3.33E: Consider the polynomials a linearly independent set in ? Why or why...
 4.3.34E: Consider the polynomials By inspection, write a linear dependence r...
 4.3.35E: Let V be a vector space that contains a linearly independent set De...
 4.3.36E: Find bases for H, K, and H C K. (See Exercises 33 and 34 in Section...
 4.3.37E: [M] Show that is a linearly independent set of functions defined on...
 4.3.38E: [M] Show that is a linearly independentset of functions defined on ...
Solutions for Chapter 4.3: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 4.3
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 37 problems in chapter 4.3 have been answered, more than 32540 students have viewed full stepbystep solutions from this chapter. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Chapter 4.3 includes 37 full stepbystep solutions.

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).

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

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

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.

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.