×
×

# Solutions for Chapter 5.5: Inequalities in One Variable

## Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition

ISBN: 9781559537636

Solutions for Chapter 5.5: Inequalities in One Variable

Solutions for Chapter 5.5
4 5 0 338 Reviews
12
2
##### ISBN: 9781559537636

This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Since 18 problems in chapter 5.5: Inequalities in One Variable have been answered, more than 9619 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. Chapter 5.5: Inequalities in One Variable includes 18 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

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.

• Diagonalization

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.

• Factorization

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.

• Fundamental Theorem.

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.

• Lucas numbers

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).

• Orthogonal subspaces.

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.

• Toeplitz matrix.

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.

×