- Appendix.1: In Exercises 14, use mathematical induction to prove that the formu...
- Appendix.2: In Exercises 14, use mathematical induction to prove that the formu...
- Appendix.3: In Exercises 14, use mathematical induction to prove that the formu...
- Appendix.4: In Exercises 14, use mathematical induction to prove that the formu...
- Appendix.5: In Exercises 5 and 6, propose a formula for the sum of the first te...
- Appendix.6: In Exercises 5 and 6, propose a formula for the sum of the first te...
- Appendix.7: In Exercises 7 and 8, use mathematical induction to prove the inequ...
- Appendix.8: In Exercises 7 and 8, use mathematical induction to prove the inequ...
- Appendix.9: Prove that for all integers
- Appendix.10: (From Chapter 2) Use mathematical induction to prove that assuming ...
- Appendix.11: (From Chapter 3) Use mathematical induction to prove that where are...
- Appendix.12: If is an integer and is odd, then is odd. (Hint: An odd number can ...
- Appendix.13: If and are real numbers and then
- Appendix.14: If and are real numbers such that and then
- Appendix.15: If and are real numbers and then
- Appendix.16: If a and b are real numbers and _a _ b_2 _ a2 _ b2, then a _ 0 or b...
- Appendix.17: If is a real number and then
- Appendix.18: Use proof by contradiction to prove that the sum of a rational numb...
- Appendix.19: (From Chapter 4) Use proof by contradiction to prove that in a give...
- Appendix.20: (From Chapter 4) Let be a linearly independent set. Use proof by co...
- Appendix.21: If and are real numbers and then
- Appendix.22: The product of two irrational numbers is irrational.
- Appendix.23: If and are real numbers such that and then
- Appendix.24: If is a polynomial function and then
- Appendix.25: If and are differentiable functions and then
- Appendix.26: (From Chapter 2) If and are matrices and then
- Appendix.27: (From Chapter 3) If is a matrix, then det_A_1_
Solutions for Chapter Appendix: Mathematical Induction and Other Forms of Proofs
Full solutions for Elementary Linear Algebra | 6th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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].
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.