- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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.
Remove row i and column j; multiply the determinant by (-I)i + j •
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(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
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 ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Solvable system Ax = b.
The right side b is in the column space of A.
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