 1.2.1: Find the mean, median, and mode for each data set. a. {1, 5, 7, 3, ...
 1.2.2: Find the mean, median, and mode for each dot plot. a. b.
 1.2.3: Students were asked how many pets they had. Their responses are sho...
 1.2.4: This graph gives the lift heights and vertical drops of the five ta...
 1.2.5: If you purchase 16 grocery items at an average cost of $1.14, what ...
 1.2.6: An ocean wave caused by an earthquake, landslide, or volcano is cal...
 1.2.7: The first three members of the stiltwalking relay team finished th...
 1.2.8: Noah scored 88, 92, 85, 65, and 89 on five tests in his history cla...
 1.2.9: At a state political rally, a speaker announced, We should raise te...
 1.2.10: This table gives information about ten of the largest saltwater fis...
 1.2.11: Create a set of data that fits each description.a.The mean age of a...
 1.2.12: This data set represents the ages of the 20 highestpaid athletes i...
 1.2.13: Fifteen students gave their ages in months. 168 163 142 163 165 164...
 1.2.14: Use this segment to measure or calculate in 14ac. a. What is the l...
Solutions for Chapter 1.2: Summarizing Data with Measures of Center
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 1.2: Summarizing Data with Measures of Center
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Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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