 2.1.1: List these fractions in increasing order by estimating their values...
 2.1.2: Ms. Lenz collecd inteformation about the students in her class.Writ...
 2.1.3: Phrases such as miles per gallon, parts per million (ppm), and acci...
 2.1.4: What number should you multiply by to solve for the unknown in each...
 2.1.5: Find the value of the unknown number in each proportion.
 2.1.6: APPLICATION Write a proportion for each problem, and solve for the ...
 2.1.7: Write three other true proportions using the four values in each pr...
 2.1.8: APPLICATION Jeremy has a job at the movie theater. His hourly wage ...
 2.1.9: In a resort area during the summer months, only one out of eight pe...
 2.1.10: APPLICATION To make three servings of Irish porridge, you need 4 cu...
 2.1.11: APPLICATION When chemists write formulas for chemical compounds, th...
 2.1.12: In the dot plot below, circle the points that represent values for ...
 2.1.13: The Forbes Celebrity 100 (www.forbes.com) listed these ten people (...
 2.1.14: Use the order of operations to evaluate these expressions. Check yo...
Solutions for Chapter 2.1: Proportions
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 2.1: Proportions
Get Full SolutionsDiscovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. Since 14 problems in chapter 2.1: Proportions have been answered, more than 7880 students have viewed full stepbystep solutions from this chapter. Chapter 2.1: Proportions includes 14 full stepbystep solutions. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

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

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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).

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

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

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.