- Chapter 1: Prerequisites
- Chapter 1.1: REAL NUMBERS: ALGEBRA ESSENTIALS
- Chapter 1.2: EXPONENTS AND SCIENTIFIC NOTATION
- Chapter 1.3: RADICALS AND RATIONAL EXPRESSIONS
- Chapter 1.4: POLYNOMIALS
- Chapter 1.5: FACTORING POLYNOMIALS
- Chapter 1.6: RATIONAL EXPRESSIONS
- Chapter 2: Equations and Inequalities
- Chapter 2.1: THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS
- Chapter 2.2: LINEAR EQUATIONS IN ONE VARIABLE
- Chapter 2.3: MODELS AND APPLICATIONS
- Chapter 2.4: COMPLEX NUMBERS
- Chapter 2.5: QUADRATIC EQUATIONS
- Chapter 2.6: OTHER TYPES OF EQUATIONS
- Chapter 2.7: LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES
- Chapter 3: Functions
- Chapter 3.1: Functions and Function Notation
- Chapter 3.2: DOMAIN AND RANGE
- Chapter 3.3: RATES OF CHANGE AND BEHAVIOR OF GRAPHS
- Chapter 3.4: COMPOSITION OF FUNCTIONS
- Chapter 3.5: TRANSFORMATION OF FUNCTIONS
- Chapter 3.6: ABSOLUTE VALUE FUNCTIONS
- Chapter 3.7: INVERSE FUNCTIONS
- Chapter 4: Linear Functions
- Chapter 4.1: LINEAR FUNCTIONS
- Chapter 4.2: MODELING WITH LINEAR FUNCTIONS
- Chapter 4.3: FITTING LINEAR MODELS TO DATA
- Chapter 5: Polynomial and Rational Functions
- Chapter 5.1: QUADRATIC FUNCTIONS
- Chapter 5.2: POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS
- Chapter 5.3: GRAPHS OF POLYNOMIAL FUNCTIONS
- Chapter 5.4: DIVIDING POLYNOMIALS
- Chapter 5.5: ZEROS OF POLYNOMIAL FUNCTIONS
- Chapter 5.6: RATIONAL FUNCTIONS
- Chapter 5.7: INVERSES AND RADICAL FUNCTIONS
- Chapter 5.8: MODELING USING VARIATION
- Chapter 6: Exponential and Logarithmic Functions
- Chapter 6.1: EXPONENTIAL FUNCTIONS
- Chapter 6.2: GRAPHS OF EXPONENTIAL FUNCTIONS
- Chapter 6.3: LOGARITHMIC FUNCTIONS
- Chapter 6.4: GRAPHS OF LOGARITHMIC FUNCTIONS
- Chapter 6.5: LOGARITHMIC PROPERTIES
- Chapter 6.6: EXPONENTIAL AND LOGARITHMIC EQUATIONS
- Chapter 6.7: EXPONENTIAL AND LOGARITHMIC MODELS
- Chapter 6.8: FITTING EXPONENTIAL MODELS TO DATA
- Chapter 7: systems Of equAtiONs ANd iNequAlities
- Chapter 7.1: SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES
- Chapter 7.2: SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES
- Chapter 7.3: SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES
- Chapter 7.4: PARTIAL FRACTIONS
- Chapter 7.5: MATRICES AND MATRIX OPERATIONS
- Chapter 7.6: SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION
- Chapter 7.7: SOLVING SYSTEMS WITH INVERSES
- Chapter 7.8: SOLVING SYSTEMS WITH CRAMER'S RULE
- Chapter 8: ANAlytiC geOmetry
- Chapter 8.1: THE ELLIPSE
- Chapter 8.2: THE HYPERBOLA
- Chapter 8.3: THE PARABOLA
- Chapter 8.4: ROTATION OF AXIS
- Chapter 8.5: CONIC SECTIONS IN POLAR COORDINATES
- Chapter 9: sequeNCes, PrObAbility ANd COuNtiNg theOry
- Chapter 9.1: SEQUENCES AND THEIR NOTATIONS
- Chapter 9.2: ARITHMETIC SEQUENCES
- Chapter 9.3: GEOMETRIC SEQUENCES
- Chapter 9.4: SERIES AND THEIR NOTATIONS
- Chapter 9.5: COUNTING PRINCIPLES
- Chapter 9.6: BINOMIAL THEOREM
- Chapter 9.7: PROBABILITY
College Algebra 1st Edition - Solutions by Chapter
Full solutions for College Algebra | 1st Edition
peA) = det(A - AI) has peA) = zero matrix.
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).
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].
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.
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.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
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.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Every v in V is orthogonal to every w in W.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.