- 7.3.1E: In, functions f and g are defined by arrow diagrams. Find g ? f and...
- 7.3.2E: In, functions f and g are defined by arrow diagrams. Find g ? f and...
- 7.3.3E: In, functions F and G are defined by formulas. Find G ? F and F ? G...
- 7.3.4E: In, functions F and G are defined by formulas. Find G ? F and F ? G...
- 7.3.5E: Define f: R ? R by the rule f(x) = ?x for all real numbers x. Find ...
- 7.3.6E: Define F: Z ? Z and G: Z ? Z by the rules F(a) = 7a and G(a) = amod...
- 7.3.7E: Define H: Z ? Z and K: Z ? Z by the rules H(a) = 6a and K(a) = amod...
- 7.3.8E: Define L: Z ? Z and M: Z ? Z by the rules L(a) = a2 and M(a) = a mo...
- 7.3.9E: The functions of each pair are inverse to each other. For each pair...
- 7.3.10E: The functions of each pair are inverse to each other. For each pair...
- 7.3.11E: The functions of each pair are inverse to each other. For each pair...
- 7.3.12E: Explain how it follows from the definition of logarithm thata. logb...
- 7.3.13E: Prove Theorem (b): If f is any function from a set X to a set Y, th...
- 7.3.14E: Prove Theorem (b): If f: X ? Y is a one-to-one and onto function wi...
- 7.3.15E: Suppose Y and Z are sets and g: Y ^ Z is a one-to-one function. Thi...
- 7.3.16E: If f: X ? Y and g: Y ? Z are functions and g ? f is one-to-one, mus...
- 7.3.17E: If f: X ? Y and g: Y ? Z are functions and g ? f is onto, must f be...
- 7.3.18E: If f: X ? Y and g: Y ? Z are functions and g ? f is one-to-one, mus...
- 7.3.19E: If f: X ? Y and g: Y ? Z are functions and g ? f is onto, must g be...
- 7.3.20E: Let f: W ? X, g: X ? Y, and h: Y ? Z be functions. Must h ? (g ? f)...
- 7.3.21E: True or False? Given any set X and given any functions f: X ? X, g:...
- 7.3.22E: True or False? Given any set X and given any functions f: X ? X, g:...
- 7.3.23E: Find g ? f, (g ? f)?1, g?1, f?1, and f?1 ? g?1, and state how (g ? ...
- 7.3.24E: Find g ? f, (g ? f)?1, g?1, f?1, and f?1 ? g?1, and state how (g ? ...
- 7.3.25E: Prove or give a counterexample: If f: X ? Y and g: Y ? X are functi...
- 7.3.26E: Suppose f: X ? Y and g: Y ? Z are both one-to-one and onto. Prove t...
- 7.3.27E: Let f: X ? Y and g: Y ? Z. Is the following property true or false?...
Solutions for Chapter 7.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications | 4th Edition
Tv = Av + Vo = linear transformation plus shift.
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Remove row i and column j; multiply the determinant by (-I)i + j •
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.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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).
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Every v in V is orthogonal to every w in W.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
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