- 5.5.1: Tell what operation on the first inequality gives the second one, a...
- 5.5.2: Find three values of the variable that satisfy each inequality. a. ...
- 5.5.3: Give the inequality graphed on each number line. a. b. c. d. e.
- 5.5.4: Translate each phrase into symbols. a. 3 is more than x b. y is at ...
- 5.5.5: Solve each equation for y. a. 3x + 4y = 5.2 b. 3(y 5) = 2x
- 5.5.6: Solve each inequality and show your work. a. 4.1 + 3.2x > 18 b. 7.2...
- 5.5.7: Solve each inequality and graph the solutions on a number line. a. ...
- 5.5.8: Ezra received $50 from his grandparents for his birthday. He makes ...
- 5.5.9: For each graph, tell what operation moves the two points in the ine...
- 5.5.10: Tell whether each inequality is true or false for the given value. ...
- 5.5.11: Solve each inequality. Explain the meaning of the result. On a numb...
- 5.5.12: Data collected by a motion sensor will vary slightly in accuracy. A...
- 5.5.13: You read the inequality symbols, <, , >, and , as is less than, is ...
- 5.5.14: The table gives equations that model the three vehicles distances i...
- 5.5.15: In Example B, the inequality 8 0.25x < 5 was written to represent t...
- 5.5.16: List the order in which you would perform these operations to get t...
- 5.5.17: The table shows the 2004 U.S. postal rates for letters, large envel...
- 5.5.18: Use the distributive property to rewrite each expression without us...
Solutions for Chapter 5.5: Inequalities in One Variable
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
peA) = det(A - AI) has peA) = zero matrix.
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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].
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-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.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
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).
Every v in V is orthogonal to every w in W.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.