Math 1B03 1ZC3 Summ T Peri x Secure www child 2phpassign 9 P

Math 1B03 1ZC3. Summ T Peri x Secure www child 2.php?assign 9 Problem #8: Which of the following sets are closed under addition? i) The set of all vectors in R of the form (a, b) where b- a (ii) The set of all 2 x 2 matrices that have the vector D-3 2]1 as an eigenvector. (iii) The set of all polynomials in P2 of the form ao al x+ a2 x2 where ao a2 (A) (ii) only GB (i) and (ii) only (C) (i) and (iii) only (D) all of them (E) (ii) and (iii) only (F) (i) only (G) (iii) only (H) none of them Problem #8 A Just Save Your work has been saved! Back to Admin Page Submit Problem #8 for Grading Problem #8 Attempt 1 Attempt #2 Attempt 3 Your Answer: Your Mark 0/2X 0/2X 34 AM Type here to search 2017-U6-10

Solution

8(i). A vector in R² of the form (a,b), where b = a is the point (a,a), where a is an arbitrary real number.The set of all such vectors (a,a) is, therefore, the set of all the points in the real x-y plane. This set is an infinite open set. Thus, thuis set cannot be a closed set.

(ii). Let A =

a

b

c

d

where a,b,c, d are arbitrary real numbers.We know that the eigenvector( X of a 2x2 matrix A is solution to the equation (A- I₂)X = 0, where is the eigenvalue of A corresponding to the eigencector (-3,2)^T. Since X = (-3,2)^T is an eigenvector of A, we have -3(a- )+2b = 0 or, 2b-3a = 3 …(1) and -3c+2(d- )=2d-3c = 2..(2).On multiplying the 1^st and the 2^ndequations by 2 and 3 respectively,we have 4b-6a =6 and           6d-9c =6 so that 4b-6a= 6d-9c or, 6a -4b-9c+6d = 0. Since this is a single equation in 4 variables a,b,c and d, it will have infinite solutions. Therefore, the number of 2x2 matrices with eigencector (-3,2)^T is infinite. Hence the set of all 2x2 matrices with eigencector (-3,2)^T cannot be a closed set.

(iii). Let P(x) = a₀+a₁x+a₂x². Now, if a₀ = a₂², then P(x) = a₂²+a₁x+a₂x². Since the number of such polymials P(x) of this form in P₂ can be infinite, hence the set of all the polynomials P(x) of the form a₂²+a₁x+a₂x² in P₂ cannot be a closed set.

Option (H) is the correct answer.

a	b
c	d