- 6.4.1E: Find the expansion of (x + y)4a) using combinatoria l reasoning, as...
- 6.4.2E: Find the expansion of (x + y)5a) using combinatorial reasoning, as ...
- 6.4.3E: Find the expansion of (x + y)6.
- 6.4.4E: Find the coefficient of x5y8 in (x + y)13.
- 6.4.5E: How many term s are there in the expansion of (x + y)100after like ...
- 6.4.6E: What is the coefficient of x7 in ( 1 + x)11?
- 6.4.7E: What is the coefficient of x9 in ( 2 ? x)19?
- 6.4.8E: What is the coefficient of x8y9 in the expansion of (3x+2y)17?
- 6.4.9E: What is the coefficient of x101y99 in the expansion of (2x ? 3y)200?
- 6.4.11E: Give a formula for the coefficient of xk in the expansion of (x2 ? ...
- 6.4.10E: Give a formula for the coefficient of xk in the expansion of (x + 1...
- 6.4.12E: The row of Pascal’s triangle containing the binomial coefficients 1...
- 6.4.15E: Show that for all positive integers n and all integers k with 0 ? k...
- 6.4.13E: What is the row of Pascal’s triangle containing the binomial coeffi...
- 6.4.14E: Show that if n is a positive integer, then
- 6.4.18E: Suppose that b is an integer with b ? 7. Use the binomial theorem a...
- 6.4.19E: Prove Pascal’s identity, using the formula for
- 6.4.16E: a) Use Exercise 14 and Corollary 1 to show that if n is an integer ...
- 6.4.17E: Show that if n and k are integers with 1 ? k ? n, then
- 6.4.21E: Prove that if n and k are integers with 1 ? k ? n, then a) using a ...
- 6.4.20E: Suppose that k and n are integers with 1 ? khexagon identity which ...
- 6.4.22E: Prove the identity whenever n, r , and k are nonnegative integers w...
- 6.4.23E: Show that if n and k are positive integers, then Use this identity ...
- 6.4.24E: Show that if p is a prime and k is an integer such that 1 ? k ? p ?...
- 6.4.25E: Let n be a positive integer. Show that
- 6.4.28E: Show that if n is a positive integer, then a) using a combinatorial...
- 6.4.26E: Let n and k be integers with 1 ? k ? n , Show that
- 6.4.29E: Give a combinatorial proof that [Hint: Count in two ways the number...
- 6.4.27E: Prove the hockeystick identity whenever n and r are positive intege...
- 6.4.30E: Give a combinatorial proof that [Hint: Count in two ways the number...
- 6.4.32E: Prove the binomial theorem using mathematical induction.
- 6.4.31E: Show that a nonempty set has the same number of subsets with an odd...
- 6.4.33E: In this exercise we will count the number of path s in the xy plane...
- 6.4.34E: Use Exercise 33 to give an alternative proof of Corollary 2 in Sect...
- 6.4.35E: Use Exercise 33 to prove Theorem 4. [Hint: Count the number of path...
- 6.4.36E: Use Exercise 33 to prove Pascal’s identity. [Hint: Show that a path...
- 6.4.37E: Use Exercise 33 to prove the hockeystick identity from Exercise 27....
- 6.4.39E: Determine a formula involving binomial coefficients for the nth ter...
- 6.4.38E: Give a combinatorial proof that if n is a positive integer then [Hi...
Solutions for Chapter 6.4: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th 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.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Remove row i and column j; multiply the determinant by (-I)i + j •
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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 = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
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.
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.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Every v in V is orthogonal to every w in W.
Outer product uv T
= column times row = rank one matrix.
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.
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.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).