 12.1: When a player hits the ball in a straightline from T to H, the path...
 12.2: The designer of the golf course decidesto make the hole more diffic...
 12.3: Write the path of the ball in as a composition of two translations....
 12.4: The designer decides to remove thebarrier and put a revolving obsta...
 12.5: Tell whether each transformation appears to be a translation1
 12.6: Tell whether each transformation appears to be a translation2
 12.7: A landscape architect represents a flower bed by a polygon with ver...
 12.8: Tell whether each transformation appears to be a rotation.1
 12.9: Tell whether each transformation appears to be a rotation.2
 12.10: Rotate the figure with the given vertices about the origin using th...
 12.11: Rotate the figure with the given vertices about the origin using th...
 12.12: Draw the result of the following compositionof transformations. Tra...
 12.13: ABC with vertices A (1, 0) , B (1, 3) , and C (2, 3) is reflected a...
 12.14: Tell whether each transformation appears to be a dilation.1
 12.15: Tell whether each transformation appears to be a dilation.2
 12.16: Tell whether each transformation appears to be a dilation.3
 12.17: Draw the image of the figure with the given vertices under a dilati...
 12.18: Draw the image of the figure with the given vertices under a dilati...
 12.19: ABCD is a square with vertices A (3, 1) ,B (3, 3) , C (1, 3), an...
 12.20: Determine the value of x if ABC BDC.Justify your answer.
 12.21: ABC is reflected across line m. a. What observations can be made ab...
 12.22: Given the coordinates of points A, B, and C,explain how you could d...
 12.23: Proving that the diagonals of rectangle KLMNare equal using a coord...
 12.24: AB has endpoints A (0, 3) and B (2, 5) . a. Draw AB and its image, ...
Solutions for Chapter 12: Extending Transformational Geometry
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 12: Extending Transformational Geometry
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Geometry, edition: 1. Chapter 12: Extending Transformational Geometry includes 24 full stepbystep solutions. Since 24 problems in chapter 12: Extending Transformational Geometry have been answered, more than 47051 students have viewed full stepbystep solutions from this chapter. Geometry was written by and is associated to the ISBN: 9780030923456.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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

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 ().

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).