 4.4.1E: In Exercises 1–4, find the vector x determined by the given coordin...
 4.4.2E: In Exercises 1–4, find the vector x determined by the given coordin...
 4.4.3E: In Exercises 1–4, find the vector x determined by the given coordin...
 4.4.4E: In Exercises 1–4, find the vector x determined by the given coordin...
 4.4.5E: In Exercises 5–8, find the coordinate vector of x relative to the g...
 4.4.6E: In Exercises 5–8, find the coordinate vector B of x relative to the...
 4.4.7E: In Exercises 5–8, find the coordinate vector of x relative to the g...
 4.4.8E: In Exercises 5–8, find the coordinate vector of x relative to the g...
 4.4.9E: In Exercises 9 and 10, find the changeofcoordinates matrix from t...
 4.4.10E: In Exercises 9 and 10, find the changeofcoordinates matrix from t...
 4.4.11E: In Exercises 11 and 12, use an inverse matrix to find B for the giv...
 4.4.12E: In Exercises 11 and 12, use an inverse matrix to find for the given...
 4.4.13E: The set B = {1 + t2, t + t2, 1 + 2t + t2} is a basis for P2.Find th...
 4.4.14E: The set B = {1  t2, t  t2, 2  t + t2} is a basis for P2.Find the...
 4.4.15E: In Exercises 15 and 16, mark each statement True or False. Justify ...
 4.4.16E: In Exercises 15 and 16, mark each statement True or False. Justify ...
 4.4.17E: The vectorsv1 = v2 = v3 = span R2but do not form a basis. Find two ...
 4.4.18E: Let B = {b1, . . . ,bn} be a basis for a vector space V. Explain wh...
 4.4.19E: Let S be a finite set in a vector space V with the property that ev...
 4.4.20E: Suppose is a linearly dependent spanning set for a vector space V. ...
 4.4.21E: Let Since the coordinate mapping determined by B is a linear transf...
 4.4.22E: Let B = {b1, . . . , bn} be a basis for Rn.Produce a description of...
 4.4.23E: Exercises 23–26 concern a vector space V, a basis and the coordinat...
 4.4.24E: Exercises 23–26 concern a vector space V, a basis and the coordinat...
 4.4.25E: Exercises 23–26 concern a vector space V, a basis and the coordinat...
 4.4.26E: Exercises 23–26 concern a vector space V, a basis and the coordinat...
 4.4.27E: In Exercises 27–30, use coordinate vectors to test the linear indep...
 4.4.28E: In Exercises 27–30, use coordinate vectors to test the linear indep...
 4.4.29E: In Exercises 27–30, use coordinate vectors to test the linear indep...
 4.4.30E: In Exercises 27–30, use coordinate vectors to test the linear indep...
 4.4.31E: Use coordinate vectors to test whether the following sets of polyno...
 4.4.32E: a. Use coordinate vectors to show that these polynomials form a bas...
 4.4.33E: In Exercises 33 and 34, determine whether the sets of polynomials f...
 4.4.34E: In Exercises 33 and 34, determine whether the sets of polynomials f...
 4.4.35E: Show that x is in H and find the coordinate vector of x, for
 4.4.36E: Show that is a basis forH and x is inH, and find the coordinate ve...
 4.4.37E: [M] Exercises 37 and 38 concern the crystal lattice for titanium, w...
 4.4.38E: [M] Exercises 37 and 38 concern the crystal lattice for titanium, w...
Solutions for Chapter 4.4: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 4.4
Get Full SolutionsLinear 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 4.4 includes 38 full stepbystep solutions. Since 38 problems in chapter 4.4 have been answered, more than 34677 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

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].

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.