- 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
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Column space C (A) =
space of all combinations of the columns of A.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = 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.
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).
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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