 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 8137 students have viewed full stepbystep solutions from this chapter.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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

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.

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

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.