 4.2.1E: Determine if is in Nul A, where
 4.2.2E: Determine if is in Nul A, where
 4.2.3E: In Exercises 3–6, find an explicit description of Nul A, by listing...
 4.2.4E: In Exercises 3–6, find an explicit description of Nul A, by listing...
 4.2.5E: In Exercises 3–6, find an explicit description of Nul A, by listing...
 4.2.6E: In Exercises 3–6, find an explicit description of Nul A, by listing...
 4.2.7E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.8E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.9E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.10E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.11E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.12E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.13E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.14E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.15E: In Exercises 15 and 16, find A such that the given set is Col A.
 4.2.16E: In Exercises 15 and 16, find A such that the given set is Col A.
 4.2.17E: For the matrices in Exercises 17–20, (a) find ksuch that Nul A is a...
 4.2.18E: For the matrices in Exercises 17–20, (a) find ksuch that Nul A is a...
 4.2.19E: For the matrices in Exercises 17–20, (a) find ksuch that Nul A is a...
 4.2.20E: For the matrices in Exercises 17–20, (a) find ksuch that Nul A is a...
 4.2.21E: With A as in Exercise 17, find a nonzero vector in Nul A and a nonz...
 4.2.22E: With A as in Exercise 18, find a nonzero vector in Nul A and a nonz...
 4.2.23E: Determine if w is in Col A. Is w in Nul A?
 4.2.24E: Determine if w is in Col A. Is w in Nul A?
 4.2.25E: In Exercises 25 and 26, A denotes an m × n matrix. Mark each statem...
 4.2.26E: In Exercises 25 and 26, A denotes an m × n matrix. Mark each statem...
 4.2.27E: It can be shown that a solution of the system below is Use this fac...
 4.2.28E: Consider the following two systems of equations: It can be shown th...
 4.2.29E: Prove Theorem 3 as follows: Given an m × n matrix A, an element in ...
 4.2.30E: Let T : V W be a linear transformation from a vector space V into a...
 4.2.31E: Define For instance, if a. Show that T is a linear transformation. ...
 4.2.32E: Define a linear transformation Find polynomials p1 and p2 in that s...
 4.2.33E: Let M2×2 be the vector space of all 2 × 2 matrices, and define a. S...
 4.2.34E: (Calculus required) Define as follows:For be the antiderivative F o...
 4.2.35E: Let V and W be vector spaces, and let T : V W be a linear transform...
 4.2.36E: Given T : V W as in Exercise 35, and given a subspace Z of W, let U...
 4.2.37E: [M] Determine whether w is in the column space of A, the null space...
 4.2.38E: [M] Determine whether w is in the column space of A, the null space...
 4.2.39E: denote the columns of the matrix A, where a. Explain why a3 and a5 ...
 4.2.40E: Then H and K are subspaces of . In fact, H and K are planes in thro...
Solutions for Chapter 4.2: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 4.2
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. Since 40 problems in chapter 4.2 have been answered, more than 35240 students have viewed full stepbystep solutions from this chapter. Chapter 4.2 includes 40 full stepbystep solutions.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

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

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

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

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.

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 •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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