 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 . What...
 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: a hyperplane [f : d] (in this case, a line) that strictly separates
 8.4.24E: Repeat Exercise 23 for Reference Exercise 23: a hyperplane [f : d] ...
 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:
 8.4.30E: Prove that the convex hull of a bounded set is bounded.
Solutions for Chapter 8.4: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 8.4
Get Full SolutionsSince 29 problems in chapter 8.4 have been answered, more than 41042 students have viewed full stepbystep solutions from this chapter. 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: 9780321982384. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 8.4 includes 29 full stepbystep solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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