 3.1: The blue line is a transversal.a. Name four pairs of corresponding ...
 3.2: Classify each pair of angles as alternate interior angles, samesid...
 3.3: Classify each pair of angles as alternate interior angles, samesid...
 3.4: Classify each pair of angles as alternate interior angles, samesid...
 3.5: Classify each pair of angles as alternate interior angles, samesid...
 3.6: Classify each pair of angles as alternate interior angles, samesid...
 3.7: Classify each pair of angles as alternate interior angles, samesid...
 3.8: Classify each pair of angles as alternate interior angles, samesid...
 3.9: Classify each pair of angles as alternate interior angles, samesid...
 3.10: Classify each pair of lines as intersecting, parallel, or skewa. AB...
 3.11: Name six lines parallel to GL.
 3.12: Name several lines skew to GL.
 3.13: Name five lines parallel to plane ABCD.
 3.14: Name two coplanar segments that donot intersect and yet are not par...
 3.15: Complete each statement with the word always, sometimes, or never. ...
 3.16: Complete each statement with the word always, sometimes, or never. ...
 3.17: Complete each statement with the word always, sometimes, or never. ...
 3.18: Complete each statement with the word always, sometimes, or never. ...
 3.19: Complete each statement with the word always, sometimes, or never. ...
 3.20: In Exercises 1820 use two lines of notebook paper as parallel line...
 3.21: Draw a large diagram showing three transversals intersectingtwo non...
 3.22: Draw a diagram of a sixsided box by following the steps below.TopB...
 3.23: Exercises 2329 refer to the diagram in Step 2 of Exercise 22. Name...
 3.24: Exercises 2329 refer to the diagram in Step 2 of Exercise 22. Name...
 3.25: Exercises 2329 refer to the diagram in Step 2 of Exercise 22. Name...
 3.26: Exercises 2329 refer to the diagram in Step 2 of Exercise 22. Name...
 3.27: Exercises 2329 refer to the diagram in Step 2 of Exercise 22. Name...
 3.28: Exercises 2329 refer to the diagram in Step 2 of Exercise 22. How ...
 3.29: Exercises 2329 refer to the diagram in Step 2 of Exercise 22. Supp...
 3.30: Complete each statement with the word always, sometimes, or never. ...
 3.31: Complete each statement with the word always, sometimes, or never. ...
 3.32: Complete each statement with the word always, sometimes, or never. ...
 3.33: Complete each statement with the word always, sometimes, or never. ...
 3.34: Complete each statement with the word always, sometimes, or never. ...
 3.35: Complete each statement with the word always, sometimes, or never. ...
 3.36: Complete each statement with the word always, sometimes, or never. ...
 3.37: Complete each statement with the word always, sometimes, or never. ...
 3.38: Complete each statement with the word always, sometimes, or never. ...
 3.39: Complete each statement with the word always, sometimes, or never. ...
 3.40: Draw each figure described. Lines a and b are skew, lines b and c a...
 3.41: Draw each figure described. Lines d and e are skew, lines e and /ar...
 3.42: Draw each figure described. Line / II plane X. plane X 'I plane Y. ...
 3.43: Complete. In an octagon the sum of the measures of the interior ang...
 3.44: Complete. In an octagon the sum of the measures of the interior ang...
 3.45: Write a two column proof. Given: WX XY;Z_ 1 is comp. to Z.3.Prove: ...
 3.46: Write a two column proof. Given: RU II ST; LR = LTProve: RS UT
Solutions for Chapter 3: Definitions
Full solutions for Geometry  1st Edition
ISBN: 9780395977279
Solutions for Chapter 3: Definitions
Get Full SolutionsGeometry was written by and is associated to the ISBN: 9780395977279. This textbook survival guide was created for the textbook: Geometry, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3: Definitions includes 46 full stepbystep solutions. Since 46 problems in chapter 3: Definitions have been answered, more than 5377 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.

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).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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 ().

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

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

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

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.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·