- 1.1.1E: Fill in the blanks using a variable or variables to rewrite the giv...
- 1.1.2E: Fill in the blanks using a variable or variables to rewrite the giv...
- 1.1.3E: Fill in the blanks using a variable or variables to rewrite the giv...
- 1.1.4E: Fill in the blanks using a variable or variables to rewrite the giv...
- 1.1.5E: Fill in the blanks using a variable or variables to rewrite the giv...
- 1.1.6E: Fill in the blanks using a variable or variables to rewrite the giv...
- 1.1.7E: Rewrite the following statements less formally, without using varia...
- 1.1.8E: Fill in the blanks to rewrite the given statement.ExerciseFor all o...
- 1.1.9E: Fill in the blanks to rewrite the given statement.ExerciseFor all e...
- 1.1.10E: Fill in the blanks to rewrite the given statement.ExerciseEvery non...
- 1.1.11E: Fill in the blanks to rewrite the given statement.ExerciseEvery pos...
- 1.1.12E: Fill in the blanks to rewrite the given statement.ExerciseThere is ...
- 1.1.13E: Fill in the blanks to rewrite the given statement.ExerciseThere is ...
Solutions for Chapter 1.1: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications | 4th 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.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column space C (A) =
space of all combinations of the columns of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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.
A sequence of steps intended to approach the desired solution.
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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