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# Solutions for Chapter 1.8: Space Geometry

## Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition

ISBN: 9781559538824

Solutions for Chapter 1.8: Space Geometry

Solutions for Chapter 1.8
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##### ISBN: 9781559538824

This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Since 30 problems in chapter 1.8: Space Geometry have been answered, more than 23460 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.8: Space Geometry includes 30 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• 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.

• Column space C (A) =

space of all combinations of the columns of A.

• Complex conjugate

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

• 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!).

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• 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.

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Multiplication Ax

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

• 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.

• Partial pivoting.

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

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

P = aaT laTa has rank l.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Rotation matrix

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

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Schwarz inequality

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

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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