- Chapter 1: Graphs, Functions, and Models
- Chapter 1.1: Introduction to Graphing
- Chapter 1.2: Functions and Graphs
- Chapter 1.3: Linear Functions, Slope, and Applications
- Chapter 1.4: Equations of Lines and Modeling
- Chapter 1.5: Linear Equations, Functions, Zeros, and Applications
- Chapter 1.6: Solving Linear Inequalities
- Chapter 2: More on Functions
- Chapter 2.1: Increasing, Decreasing, and Piecewise Functions; Applications
- Chapter 2.2: The Algebra of Functions
- Chapter 2.3: The Composition of Functions
- Chapter 2.4: Symmetry
- Chapter 2.5: Transformations
- Chapter 2.6: Variation and Applications
- Chapter 3: Quadratic Functions and Equations; Inequalities
- Chapter 3.1: The Complex Numbers
- Chapter 3.2: Quadratic Equations, Functions, Zeros, and Models
- Chapter 3.3: Analyzing Graphs of Quadratic Functions
- Chapter 3.4: Solving Rational Equations and Radical Equations
- Chapter 3.5: Solving Equations and Inequalities with Absolute Value
- Chapter 4: Polynomial Functions and Rational Functions
- Chapter 4.1: Polynomial Functions and Modeling
- Chapter 4.2: Graphing Polynomial Functions
- Chapter 4.3: Polynomial Division; The Remainder Theorem and the Factor Theorem
- Chapter 4.4: Theorems about Zeros of Polynomial Functions
- Chapter 4.5: Rational Functions
- Chapter 4.6: Polynomial Inequalities and Rational Inequalities
- Chapter 5: Exponential Functions and Logarithmic Functions
- Chapter 5.1: Inverse Functions
- Chapter 5.2: Exponential Functions and Graphs
- Chapter 5.3: Logarithmic Functions and Graphs
- Chapter 5.4: Properties of Logarithmic Functions
- Chapter 5.5: Solving Exponential Equations and Logarithmic Equations
- Chapter 5.6: Applications and Models: Growth and Decay; Compound Interest
- Chapter 6: Systems of Equations and Matrices
- Chapter 6.1: Systems of Equations in Two Variables
- Chapter 6.2: Systems of Equations in Three Variables
- Chapter 6.3: Matrices and Systems of Equations
- Chapter 6.4: Matrix Operations
- Chapter 6.5: Inverses of Matrices
- Chapter 6.6: Determinants and Cramers Rule
- Chapter 6.7: Systems of Inequalities and Linear Programming
- Chapter 6.8: Partial Fractions
- Chapter 7: Conic Sections
- Chapter 7.1: The Parabola
- Chapter 7.2: The Circle and the Ellipse
- Chapter 7.3: The Hyperbola
- Chapter 7.4: Nonlinear Systems of Equations and Inequalities
- Chapter 8: Sequences, Series, and Combinatorics
- Chapter 8.1: Sequences and Series
- Chapter 8.2: Arithmetic Sequences and Series
- Chapter 8.3: Geometric Sequences and Series
- Chapter 8.4: Mathematical Induction
- Chapter 8.5: Combinatorics: Permutations
- Chapter 8.6: Combinatorics: Combinations
- Chapter 8.7: The Binomial Theorem
- Chapter 8.8: Probability
- Chapter R: Basic Concepts of Algebra
- Chapter R.1: The Real-Number System
- Chapter R.2: Integer Exponents, Scientific Notation, and Order of Operations
- Chapter R.3: Addition, Subtraction, and Multiplication of Polynomials
- Chapter R.4: Factoring
- Chapter R.5: The Basics of Equation Solving
- Chapter R.6: Rational Expressions
- Chapter R.7: Radical Notation and Rational Exponents
College Algebra: Graphs and Models 5th Edition - Solutions by Chapter
Full solutions for College Algebra: Graphs and Models | 5th Edition
Tv = Av + Vo = linear transformation plus shift.
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.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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.
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
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.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.