×
Get Full Access to Math - Textbook Survival Guide
Get Full Access to Math - Textbook Survival Guide
×

# Solutions for Chapter 12: Prisms

## Full solutions for Geometry | 1st Edition

ISBN: 9780395977279

Solutions for Chapter 12: Prisms

Solutions for Chapter 12
4 5 0 307 Reviews
22
2
##### ISBN: 9780395977279

Geometry was written by and is associated to the ISBN: 9780395977279. This textbook survival guide was created for the textbook: Geometry, edition: 1. Since 40 problems in chapter 12: Prisms have been answered, more than 5395 students have viewed full step-by-step solutions from this chapter. Chapter 12: Prisms includes 40 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• 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).

• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• 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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.

• 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.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Hankel matrix H.

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

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Length II x II.

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

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Network.

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

• Normal matrix.

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

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Particular solution x p.

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

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

×