- 7-6.1: Vocabulary Apply the vocabulary from this lesson to answer each que...
- 7-6.2: Vocabulary Apply the vocabulary from this lesson to answer each que...
- 7-6.3: Graphic Design A designer created this logo for a real estate agent...
- 7-6.4: Given that AOB COD, find the coordinates of C and the scale factor.
- 7-6.5: Given that ROS POQ,find the coordinates of S andthe scale factor.
- 7-6.6: Given: A (0, 0) , B (-1, 1) , C (3, 2) , D (-2, 2) , and E (6, 4)Pr...
- 7-6.7: Given: J (-1, 0) , K (-3, -4) , L (3, -2) , M (-4, -6) , and N (5, ...
- 7-6.8: Multi-Step Graph the image of each triangle after a dilation with t...
- 7-6.9: Multi-Step Graph the image of each triangle after a dilation with t...
- 7-6.10: Advertising A promoter produced this design for a street festival. ...
- 7-6.11: Given that UOV XOY, find the coordinates of X and the scale factor
- 7-6.12: Given that MON KOL, find the coordinates of K and the scale factor.
- 7-6.13: Given: D (-1, 3) , E (-3, -1) , F (3, -1) , G (-4, -3) , and H (5, ...
- 7-6.14: Given: M (0, 10) , N (5, 0) , P (15, 15) , Q (10, -10) , and R (30,...
- 7-6.15: Multi-Step Graph the image of each triangle after a dilation with t...
- 7-6.16: Multi-Step Graph the image of each triangle after a dilation with t...
- 7-6.17: Critical Thinking Consider the transformation given by the mapping(...
- 7-6.18: /////ERROR ANALYSIS///// Which solution to find the scale factor of...
- 7-6.19: Write About It A dilation maps ABC to A'B 'C '. How is the scale fa...
- 7-6.20: This problem will prepare you for the Concept Connection on page 50...
- 7-6.21: Which coordinates for C make COD similar to AOB? (0, 2.4) (0, 3)(0,...
- 7-6.22: A dilation with scale factor 2 maps RST to R'S'T'. The perimeter of...
- 7-6.23: Which triangle with vertices D, E, and F is similar to ABC?D (1, 2)...
- 7-6.24: Gridded ResonseAB with endpoints A (3, 2) and B (7, 5) is dilated b...
- 7-6.25: How many different triangles having XY as a sideare similar to MNP?
- 7-6.26: XYZ MPN. Find the coordinates of Z.
- 7-6.27: A rectangle has two of its sides on the x- andy-axes, a vertex at t...
- 7-6.28: ABC has vertices A (0, 1) , B (3, 1) , and C (1, 3) . DEF has verti...
- 7-6.29: Write an inequality to represent the situation. (Previous course)A ...
- 7-6.30: Find the length of each segment, given that DE FE .(Lesson 5-2)HF
- 7-6.31: Find the length of each segment, given that DE FE .(Lesson 5-2)JF
- 7-6.32: Find the length of each segment, given that DE FE .(Lesson 5-2)CF
- 7-6.33: SUV SRT. Find the length of each segment. (Lesson 7-4) RT
- 7-6.34: SUV SRT. Find the length of each segment. (Lesson 7-4) V T
- 7-6.35: SUV SRT. Find the length of each segment. (Lesson 7-4) ST
Solutions for Chapter 7-6: Dilations and Similarity in the Coordinate Plane
Full solutions for Geometry | 1st Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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.
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
A symmetric matrix with eigenvalues of both signs (+ and - ).
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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 .
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Solvable system Ax = b.
The right side b is in the column space of A.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).