- Chapter 1: AN INTRODUCTION TO DATA AND FUNCTIONS
- Chapter 2: RATES OF CHANGE AND LINEAR FUNCTIONS
- Chapter 3: WHEN LINES MEET: LINEAR SYSTEMS
- Chapter 4: THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE
- Chapter 5: GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS
- Chapter 6: LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS
- Chapter 7: POWER FUNCTIONS
- Chapter 8: QUADRATICS AND THE MATHEMATICS OF MOTION
- Chapter 9: NEW FUNCTIONS FROM OLD
Explorations in College Algebra 5th Edition - Solutions by Chapter
Full solutions for Explorations in College Algebra | 5th Edition
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.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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].
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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.
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.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.