 44.1: Write the translation matrix.
 44.2: Find the coordinates of ABC.
 44.3: Graph the preimage and the image.
 44.4: STANDARDIZED TEST PRACTICE A point is translated from B to C as sho...
 44.5: Write the coordinates in a vertex matrix.
 44.6: Find the coordinates of the image after a dilation by a scale facto...
 44.7: Find the coordinates of the image after a dilation by a scale facto...
 44.8: Find the coordinates of the image after a reflection over the xaxis.
 44.9: Find the coordinates of the image after a reflection over the yaxis.
 44.10: Find the coordinates of the image after a rotation of 180.
 44.11: Find the coordinates of the image after a rotation of 270.
 44.12: DEF with D(1, 4), E(2, 5), and F(6, 6), translated 4 units left ...
 44.13: MNO with M(7, 6), N(1, 7), and O(3, 1), translated 2 units right ...
 44.14: Rectangle RSUT with vertices R(3, 2), S(1, 2), U(1, 1), T(3, 1)...
 44.15: Triangle DEF with vertices D(2, 2), E(3, 5), and F(5, 2) is trans...
 44.16: ABC with A(0, 2), B(1.5, 1.5), and C(2.5, 0) is dilated so that i...
 44.17: XYZ with X(6, 2), Y(4, 8), and Z(2, 6) is dilated so that its per...
 44.18: The vertices of XYZ are X(1, 1), Y(2, 4), and Z( 7, 1). The tria...
 44.19: The vertices of rectangle ABDC are A(3, 5), B(5, 5), D(5, 1), and...
 44.20: Parallelogram DEFG with D(2, 4), E(5, 4), F(4, 1), and G(1, 1) is r...
 44.21: MNO with M(2, 6), N(1, 4), and O(3, 4) is rotated 180 counterclo...
 44.22: Write the vertex matrix. Multiply the vertex matrix by 1.
 44.23: Graph the preimage and image.
 44.24: What type of transformation does the graph represent?
 44.25: A triangle is rotated 90 counterclockwise about the origin. The coo...
 44.26: A triangle is rotated 90 clockwise about the origin. The coordinate...
 44.27: A quadrilateral is reflected across the yaxis. The coordinates of ...
 44.28: Find the coordinates of the image in matrix form after a reflection...
 44.29: Find the coordinates of the image in matrix form after a 180 rotati...
 44.30: Find the coordinates of the image in matrix form after a reflection...
 44.31: What do you observe about these three matrices? Explain.
 44.32: Write a translation matrix that can be used to move the cursor 3 in...
 44.33: If the cursor is currently at (3.5, 2.25), what are the coordinates...
 44.34: Determine the coordinates for the vertices of the fountain.
 44.35: The center of the fountain was at (5, 3.5). What will be the coor...
 44.36: GYMNASTICS The drawing at the right shows four positions of a man p...
 44.37: Describe the reflection and transformation combination shown at the...
 44.38: Write two matrix operations that can be used to find the coordinate...
 44.39: Does it matter which operation you do first? Explain.
 44.40: What are the coordinates of the next two footprints?
 44.41: Write the translation matrix for ABC and its image ABC shown at the...
 44.42: Compare and contrast the size and shape of the preimage and image f...
 44.43: OPEN ENDED Write a translation matrix that moves DEF up and left.
 44.44: CHALLENGE Do you think a matrix exists that would represent a refle...
 44.45: REASONING Determine whether the following statement is sometimes, a...
 44.46: Writing in Math Use the information about computer animation on pag...
 44.47: ACT/SAT Triangle ABC has vertices with coordinates A(4, 2), B(4, ...
 44.48: REVIEW Melanie wanted to find 5 consecutive whole numbers that add ...
 44.49: A 2 3 B 3 2 5
 44.50: A 4 1 B 2 1 51
 44.51: A 2 5 B 5 5 P
 44.52: 2 4 6 12 9 11 10 8 2 3 + 3 1 2 3 2 3 4 3 4 5
 44.53: 4 3 6 3 4 9 1 7 2 3  8 7 2 6 10 1 4 1 5
 44.54: (3, 5), (4, 6), (5, 4)
 44.55: x = 5y + 2
 44.56: x = y 2
 44.57: 5 4 3 2 1 0 1 2 3 4 5
 44.58: 6 5 4 3 2 1 0 1 2 3 4
 44.59: BUSINESS Reliable Rentals rents cars for $12.95 per day plus 15 per...
 44.60: x 8 = _3 4
 44.61: 4 20 = _1 m
 44.62: 2 3 = _a 42
 44.63: 2 y = _8 9
 44.64: 4 n = _6 2n  3
 44.65: x 5 = _x + 1 8
Solutions for Chapter 44: Transformations with Matrices
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 44: Transformations with Matrices
Get Full SolutionsAlgebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Chapter 44: Transformations with Matrices includes 65 full stepbystep solutions. Since 65 problems in chapter 44: Transformations with Matrices have been answered, more than 56429 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1.

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

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(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 .

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).

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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