- A.1.21E: Define a relation ~ on the set of ordered pairs of positive integer...
- A.1.1E: Prove Theorem 3, which states that the multiplicative identity elem...
- A.1.2E: Prove Theorem 4, which states that for every nonzero real number x,...
- A.1.3E: Prove that for all real numbers x and y, (–x)?y = x?(– y) = –(x ? y).
- A.1.4E: Prove that for all real numbers x and y, – (x + y) = (– x) + (– y).
- A.1.5E: Prove that for all real numbers x and y, (–x) ? (–y) = x ? y.
- A.1.6E: Prove that for all real numbers x, y, and z, if x + z = y + z, then...
- A.1.8E: Prove that for all real numbers x and y, x = y if and only if.x – y...
- A.1.7E: Prove that for every real number x, –(–x) = x.Define the difference...
- A.1.9E: Prove that for all real numbers x and y, – x – y = –(x + y).
- A.1.11E: Prove that for all real numbers w, x, y, and z, if x ? 0 and z ? 0,...
- A.1.10E: Prove that for all nonzero real numbers x and y, l/(x/y) = y/x. whe...
- A.1.12E: Prove that for every positive real number x, 1/x is also a positive...
- A.1.13E: Prove that for all positive real numbers x and y, x ? y is also a p...
- A.1.14E: Prove that for all real numbers x and y, if x > 0 and y < 0, then x...
- A.1.15E: Prove that for all real numbers x, y, and z, if x > y and z < 0, th...
- A.1.16E: Prove that for every real number x, x ? 0 if and only if x2 > 0.
- A.1.17E: Prove that for all real numbers w x, y, and z, if w < x and y < z, ...
- A.1.18E: Prove that for all positive real numbers x and y, if x < y, then 1/...
- A.1.19E: Prove that for every positive real number x, there exists a positiv...
- A.1.20E: Prove that between every two distinct real numbers there is a ratio...
- A.1.22E: Define a relation ? on ordered pairs of integers with second entry ...
Solutions for Chapter A.1: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
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.
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.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
= Xl (column 1) + ... + xn(column n) = combination of columns.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
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.
Solvable system Ax = b.
The right side b is in the column space of A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).