 4.4.1: Triangle ABC with vertices A(1, 4), B(2, 5), and C(6, 6) is tran...
 4.4.2: Triangle ABC with vertices A(1, 4), B(2, 5), and C(6, 6) is tran...
 4.4.3: Triangle ABC with vertices A(1, 4), B(2, 5), and C(6, 6) is tran...
 4.4.4: A point is translated from B to C as shown at the right. If a point...
 4.4.5: Write the coordinates in a vertex matri
 4.4.6: Find the coordinates of the image after a dilation by a scale facto...
 4.4.7: Find the coordinates of the image after a dilation by a scale facto...
 4.4.8: Find the coordinates of the image after a reflection over the xaxis.
 4.4.9: Find the coordinates of the image after a reflection over the yaxis.
 4.4.10: Find the coordinates of the image after a rotation of 180
 4.4.11: Find the coordinates of the image after a rotation of 270.
 4.4.12: Write the translation matrix for each figure. Then find the coordin...
 4.4.13: Write the translation matrix for each figure. Then find the coordin...
 4.4.14: Write the translation matrix for each figure. Then find the coordin...
 4.4.15: Write the translation matrix for each figure. Then find the coordin...
 4.4.16: Write the vertex matrix for each figure. Then find the coordinates ...
 4.4.17: Write the vertex matrix for each figure. Then find the coordinates ...
 4.4.18: Write the vertex matrix and the reflection matrix for each figure. ...
 4.4.19: Write the vertex matrix and the reflection matrix for each figure. ...
 4.4.20: Write the vertex matrix and the rotation matrix for each figure. Th...
 4.4.21: Write the vertex matrix and the rotation matrix for each figure. Th...
 4.4.22: Write the vertex matrix. Multiply the vertex matrix by 1
 4.4.23: Graph the preimage and image
 4.4.24: What type of transformation does the graph represent?
 4.4.25: A triangle is rotated 90 counterclockwise about the origin. The coo...
 4.4.26: A triangle is rotated 90 clockwise about the origin. The coordinate...
 4.4.27: A quadrilateral is reflected across the yaxis. The coordinates of ...
 4.4.28: Find the coordinates of the image in matrix form after a reflection...
 4.4.29: Find the coordinates of the image in matrix form after a 180 rotati...
 4.4.30: Find the coordinates of the image in matrix form after a reflection...
 4.4.31: What do you observe about these three matrices? Explain.
 4.4.32: Write a translation matrix that can be used to move the cursor 3 in...
 4.4.33: If the cursor is currently at (3.5, 2.25), what are the coordinates...
 4.4.34: Determine the coordinates for the vertices of the fountain.
 4.4.35: The center of the fountain was at (5, 3.5). What will be the coor...
 4.4.36: The drawing at the right shows four positions of a man performing t...
 4.4.37: Describe the reflection and transformation combination shown at the...
 4.4.38: Write two matrix operations that can be used to find the coordinate...
 4.4.39: Does it matter which operation you do first? Explain
 4.4.40: What are the coordinates of the next two footprints?
 4.4.41: Write the translation matrix for ABC and its image ABC shown at the...
 4.4.42: Compare and contrast the size and shape of the preimage and image f...
 4.4.43: Write a translation matrix that moves DEF up and left.
 4.4.44: Do you think a matrix exists that would represent a reflection over...
 4.4.45: Determine whether the following statement is sometimes, always, or ...
 4.4.46: Use the information about computer animation on page 185 to explain...
 4.4.47: Triangle ABC has vertices with coordinates A(4, 2), B(4, 3), and...
 4.4.48: Triangle ABC has vertices with coordinates A(4, 2), B(4, 3), and...
 4.4.49: Determine whether each matrix product is defined. If so, state the ...
 4.4.50: Determine whether each matrix product is defined. If so, state the ...
 4.4.51: Determine whether each matrix product is defined. If so, state the ...
 4.4.52: Perform the indicated matrix operations. If the matrix does not exi...
 4.4.53: Perform the indicated matrix operations. If the matrix does not exi...
 4.4.54: Graph each relation or equation and find the domain and range. Then...
 4.4.55: Graph each relation or equation and find the domain and range. Then...
 4.4.56: Graph each relation or equation and find the domain and range. Then...
 4.4.57: Write an absolute value inequality for each graph.
 4.4.58: Write an absolute value inequality for each graph.
 4.4.59: Reliable Rentals rents cars for $12.95 per day plus 15 per mile. Lu...
 4.4.60: Use cross products to solve each proportion.
 4.4.61: Use cross products to solve each proportion.
 4.4.62: Use cross products to solve each proportion.
 4.4.63: Use cross products to solve each proportion.
 4.4.64: Use cross products to solve each proportion.
 4.4.65: Use cross products to solve each proportion.
Solutions for Chapter 4.4: Transformations with Matrices
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 4.4: Transformations with Matrices
Get Full SolutionsThis textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. Since 65 problems in chapter 4.4: Transformations with Matrices have been answered, more than 44275 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.4: Transformations with Matrices includes 65 full stepbystep solutions. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568.

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

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.