(10pts) 3. Use direct proof to show that if x and y are positive real numbers,...
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers, = E(X), μγ E (Y),咳= Var(X) and e ranclom variables. with y a) show Cov(aX +b, cY +d)- ac Cov(X,Y) (b) show Corr(aX + b, cY + d)-PXY for a > 0 and c > 0.
6. Suppose that {x,] is a sequence of positive numbers and limA = a Show that if L> 1 then lim x =00, and if L < 1 lim x = 0 n+02 b. Construct a sequence of positive numbers {x,} such that lim * = 1 and the sequence {x} diverges. c. Let k E N and a > 1 Show that lim = 0. O LIVE
{x_n} and {y_n} are sequences of positive real numbers
AC fn→oo > O, prove tha m in yn lim xn 0 implies lim yn_0
Let f(x, y) 2e-(x+y), x > 0, y > 0. Show that X, Y are independent. What are the marginal PDFS of each?
Let f(x,y) = 12e-2(x+y), x > 0, y > 0. Show that X, Y are independent. What are the marginal PDFs of each?
Show the starting diene and dienophile you could use to prepare the following molecule: Diene Dienophile>
5. (10 pts) Use the inner product < x,y > = 22191 +2242 in R2 and the Gram - Schmidt process to transform {(2, -1), (-2, 10)} into an orthonormal basis
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
If a and b are real numbers and 1 < a <b, then a-1 > b-1. Proof by contradiction.