- 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
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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).
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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.
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.
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.
A sequence of steps intended to approach the desired solution.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
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.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·