- Chapter 1: An Introduction to Algebra
- Chapter 10: Quadratic Equations, Functions, and Inequalities
- Chapter 11: Exponential and Logarithmic Functions
- Chapter 12: More on Systems of Equations
- Chapter 13: Conic Sections; More Graphing
- Chapter 14: Miscellaneous Topics
- Chapter 2: Equations, Inequalities, and Problem Solving
- Chapter 3: Graphing Linear Equations and Inequalities in Two Variables; Functions
- Chapter 4: Systems of Linear Equations and Inequalities
- Chapter 5: Exponents and Polynomials
- Chapter 6: Factoring and Quadratic Equations
- Chapter 7: Rational Expressions and Equations
- Chapter 8: Transition to Intermediate Algebra
- Chapter 9: Radical Expressions and Equations
Elementary and Intermediate Algebra 5th Edition - Solutions by Chapter
Full solutions for Elementary and Intermediate 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.
Column space C (A) =
space of all combinations of the columns of A.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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.
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.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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.
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.
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.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Constant down each diagonal = time-invariant (shift-invariant) filter.