 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
ISBN: 9780618783762
Solutions for Chapter Appendix: Mathematical Induction and Other Forms of Proofs
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. Chapter Appendix: Mathematical Induction and Other Forms of Proofs includes 27 full stepbystep solutions. Since 27 problems in chapter Appendix: Mathematical Induction and Other Forms of Proofs have been answered, more than 31338 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Covariance matrix:E.
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].

Cyclic shift
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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + 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.

Multiplier eij.
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 ].

Schwarz inequality
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}.