 7.8.1: In Exercises 1 and 2, identify the basic tessellation grid (kites o...
 7.8.2: In Exercises 1 and 2, identify the basic tessellation grid (kites o...
 7.8.3: In Exercises 3 and 4, copy the figure and the grid onto patty paper...
 7.8.4: In Exercises 3 and 4, copy the figure and the grid onto patty paper...
 7.8.5: Create a glidereflection tiling design of recognizable shapes by u...
 7.8.6: Create a glidereflection tiling design of recognizable shapes by u...
 7.8.7: Find the coordinates of the circumcenter and orthocenter of FAN wit...
 7.8.8: Construction Construct a circle and a chord of the circle. With com...
 7.8.9: Remys friends are pulling him on a sled. One of his friends is stro...
 7.8.10: The green prism below right was built from the two solids below lef...
 7.8.11: For Exercises 112, identify each statement as true or false. For ea...
 7.8.12: For Exercises 112, identify each statement as true or false. For ea...
 7.8.13: In Exercises 1315, identify the type or types of symmetry, includin...
 7.8.14: In Exercises 1315, identify the type or types of symmetry, includin...
 7.8.15: In Exercises 1315, identify the type or types of symmetry, includin...
 7.8.16: The faade of Chartres Cathedral in France does not have reflectiona...
 7.8.17: Find or create a logo that has reflectional symmetry. Sketch the lo...
 7.8.18: Find or create a logo that has rotational symmetry, but not reflect...
 7.8.19: In Exercises 19 and 20, classify the tessellation and give the vert...
 7.8.20: In Exercises 19 and 20, classify the tessellation and give the vert...
 7.8.21: Experiment with a mirror to find the smallest vertical portion ( y)...
 7.8.22: Miniaturegolf pro Sandy Trapp wishes to impress her fans with a ho...
 7.8.23: In Exercises 2325, identify the shape of the tessellation grid and ...
 7.8.24: In Exercises 2325, identify the shape of the tessellation grid and ...
 7.8.25: In Exercises 2325, identify the shape of the tessellation grid and ...
 7.8.26: In Exercises 26 and 27, copy the figure and grid onto patty paper. ...
 7.8.27: In Exercises 26 and 27, copy the figure and grid onto patty paper. ...
 7.8.28: In his woodcut Day and Night, Escher gradually changes the shape of...
Solutions for Chapter 7.8: Tessellations That Use Glide Reflections
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 7.8: Tessellations That Use Glide Reflections
Get Full SolutionsDiscovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Since 28 problems in chapter 7.8: Tessellations That Use Glide Reflections have been answered, more than 22115 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.8: Tessellations That Use Glide Reflections includes 28 full stepbystep solutions.

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

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = 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 nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Outer product uv T
= column times row = rank one matrix.

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 = Q1 = Q.

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

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

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·