 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 Patricia and is associated to the ISBN: 9781559537636. Since 14 problems in chapter 2.1: Proportions have been answered, more than 2889 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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

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

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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