 6.2.1: Consider R 2 with the inner product of this section, ({x1, Xz), {yl...
 6.2.2: Consider R2 with the inner product ({xh Xz), (yh Y 2 )) = 4X1Y1 + 9...
 6.2.3: Consider R2 with the inner product ({xh Xz), (yh Y 2 )) = X1Y1 + l ...
 6.2.4: Determine the inner product that must be placed on R2for the equati...
 6.2.5: Prove that the "pseudo" inner product of Minkowski geometry violate...
 6.2.6: Use the definition of distance between two points in Minkowski spac...
 6.2.7: Determine the distance between points P(O, 0, 0, 0) and M ( 1, 0, 0...
 6.2.8: Prove that the vectors (2, 0, 0, 1) and (1, 0, 0, 2) are orthogonal...
 6.2.9: Determine the equations of the circles with center the origin and r...
 6.2.10: The star Sirius is 8 light years from Earth. Sirius is the nearest ...
 6.2.11: A spaceship makes a round trip to the bright star Capella, which is...
 6.2.12: The star cluster Pleiades in the constellation Taurus is 410 light ...
 6.2.13: The star cluster Praesepe in the constellation Cancer is 515 light ...
Solutions for Chapter 6.2: NonEuclidean Geometry and Special Relativity
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 6.2: NonEuclidean Geometry and Special Relativity
Get Full SolutionsChapter 6.2: NonEuclidean Geometry and Special Relativity includes 13 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. Since 13 problems in chapter 6.2: NonEuclidean Geometry and Special Relativity have been answered, more than 25276 students have viewed full stepbystep solutions from this chapter.

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

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.