(1 [xii) Ic: aix has two distinct real eigenvalues if and only if k >
(1 point) The matrix A = 1-6 1 8 k] 4 has two distinct real eigenvalues if and only if k > 24.5
(1 point) The matrix -1 -1 01 A = -16 0 Lk 0 has three distinct real eigenvalues if and only if -17.036 <k< 29.184
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,...
l. Let wn > 0 and 〈 > 0. Show that s2 + 2(Wns+uậ = 0 has (a) complex roots when 0 < £1. (b) real and equal roots when ς-1, and real and distinct roots when ς > 1
If a and b are real numbers and 1 < a <b, then a-1 > b-1. Proof by contradiction.
Let s = {k=1CkXAz be a simple function, where {A1, A2, ... , An} are disjoint. Prove that for every p>0, |CK|PXAR
5. Let S = {vi, u2, , v) be a set of k vectors in Rn with k > n. Show that S cannot be a basis for Rn.
3. Evaluate the product lin=1(4k/2). Prove your answer. 4. Give an asymptotically tight bound for Ση=1 kr where r > 0 is a constant.
(1 point) The matrix 2 -1 0 A2-3 -1 has three distinct real eigenvalues if and only if (1 point) The matrix 2 -1 0 A2-3 -1 has three distinct real eigenvalues if and only if
Problem 4. The median of a PDF fx(x) is defined as the number a for which P(X s a)-P(X > a)-1/2. Find the median of a Gaussian PDF N(μ; σ2).