 7.3.1: Name the single translation vector that can replace the composition...
 7.3.2: Name the single rotation that can replace the composition of these ...
 7.3.3: Lines m and n are parallel and 10 cm apart. a. Point A is 6 cm from...
 7.3.4: Two lines m and n intersect at point P, forming a 40 angle. a. You ...
 7.3.5: Copy the figure and PAL onto patty paper. Reflect the figure across...
 7.3.6: Copy the figure and the pair of parallel lines onto patty paper. Re...
 7.3.7: Copy the hexagonal figure and its translated image onto patty paper...
 7.3.8: Copy the original figure and its rotated image onto patty paper. Fi...
 7.3.9: Copy the two figures below onto graph paper. Each figure is the gli...
 7.3.10: Have you noticed that some letters have both horizontal and vertica...
 7.3.11: What combination of transformations changed the figure into the ima...
 7.3.12: In Exercises 12 and 13, sketch the next two figures
 7.3.13: In Exercises 12 and 13, sketch the next two figures
 7.3.14: If you draw a figure on an uninflated balloon and then blow up the ...
 7.3.15: List two objects in your home that have rotational symmetry, but no...
 7.3.16: Is it possible for a triangle to have exactly one line of symmetry?...
 7.3.17: Draw two points onto a piece of paper and connect them with a curve...
 7.3.18: Study these two examples of matrix addition, and then use your indu...
Solutions for Chapter 7.3: Compositions of Transformations
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 7.3: Compositions of Transformations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Since 18 problems in chapter 7.3: Compositions of Transformations have been answered, more than 22438 students have viewed full stepbystep solutions from this chapter. Chapter 7.3: Compositions of Transformations includes 18 full stepbystep solutions. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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

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·