 8.1.1: Name the coordinates of the vertices of each figure.
 8.1.2: The xcoordinates of the vertices of a triangle are entered into li...
 8.1.3: The red triangle at right is the image of the black triangle after ...
 8.1.4: Consider the square at right. a. Sketch the image of the figure aft...
 8.1.5: The spider in the upper left has its xcoordinates in list L1 and i...
 8.1.6: Consider the triangle on the calculator screen at right.a. Describe...
 8.1.7: The coordinates of the vertices of a triangle are (2, 1), (4, 3), a...
 8.1.8: APPLICATION Lisa is designing a computer animation. She has a set o...
 8.1.9: APPLICATION Nick is also designing a computer animation program. Hi...
 8.1.10: Complete this table of values for g(x) =  x 3  and h(x) = (x 3)2.
 8.1.11: Use f (x) = 2 + 3x to find a. f (5) b. x when f (x) = 10 c. f (x + ...
 8.1.12: Solve each equation. a. 5 = 3 + 2x b. 4 = 8 + 3(x 2) c. 7 + 2x = 3 + x
 8.1.13: Find an equation for each graph.
Solutions for Chapter 8.1: Translating Points
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 8.1: Translating Points
Get Full SolutionsSince 13 problems in chapter 8.1: Translating Points have been answered, more than 9005 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.1: Translating Points includes 13 full stepbystep solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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