 8.4.1E: Let L be the line in R2 through the points Find a linear functional...
 8.4.2E: Let L be the line in R2 through the points Find a linear functional...
 8.4.3E: In Exercises 3 and 4, determine whether each set is open or closed ...
 8.4.4E: In Exercises 3 and 4, determine whether each set is open or closed ...
 8.4.5E: In Exercises 5 and 6, determine whether or not each set is compact ...
 8.4.6E: In Exercises 5 and 6, determine whether or not each set is compact ...
 8.4.7E: In Exercises 7–10, let H be the hyperplane through the listed point...
 8.4.8E: In Exercises 7–10, let H be the hyperplane through the listed point...
 8.4.9E: In Exercises 7–10, let H be the hyperplane through the listed point...
 8.4.10E: In Exercises 7–10, let H be the hyperplane through the listed point...
 8.4.11E: and let H be the hyperplane in R4 with normal n and passing through...
 8.4.12E: with normal n that separates A and B. Is there a hyperplane paralle...
 8.4.13E:
 8.4.14E: Let F1 and F2 be 4dimensional flats in R6, and suppose that F1 F2 ...
 8.4.15E: In Exercises 15–20, write a formula for a linear functional f and s...
 8.4.16E: In Exercises 15–20, write a formula for a linear functional f and s...
 8.4.17E: In Exercises 15–20, write a formula for a linear functional f and s...
 8.4.18E: In Exercises 15–20, write a formula for a linear functional f and s...
 8.4.19E: In Exercises 15–20, write a formula for a linear functional f and s...
 8.4.20E: In Exercises 15–20, write a formula for a linear functional f and s...
 8.4.21E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 8.4.22E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 8.4.23E: Let v1 = v2 = v3 = and p = Finda hyperplane [f : d] (in this case, ...
 8.4.24E: Repeat Exercise 23 forv1 = v2 = v3 = and p = Reference Exercise 23:...
 8.4.25E:
 8.4.27E: Give an example of a closed subset S of R2 such that conv S is not ...
 8.4.28E: Give an example of a compact set A and a closed set B in R2 such th...
 8.4.29E: Prove that the open ball is a convex set. [Hint: Use the Triangle I...
 8.4.30E: Prove that the convex hull of a bounded set is bounded.
Solutions for Chapter 8.4: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 8.4
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.4 includes 29 full stepbystep solutions. Since 29 problems in chapter 8.4 have been answered, more than 87679 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.

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).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Outer product uv T
= column times row = rank one matrix.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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!

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.