Which of the following sets are closed under addition The se

Which of the following sets are closed under addition? The set of all vectors in R^2 of the form (a, b) where b = a^2. The set of all 3 times 3 matrices that have the vector [2 1 4]^T as an eigenvector. The set of all polynomials in Pi of the form ao + a1 x + a2 x^2 where ao = a2. none of them all of them (iii) only (ii) only (i) and (ii) only (i) and (iii) only (ii) and (iii) only (i) only

Solution

A set X is closed under addition if x+yX for any x,yX

i) the set of all vectors in R²  of the form (a,b) where b = a²

now x = (1,1), y = (2,4)    set ,but x +y = (3,5) does not not belong to set.

ii)the set of 3 *3 matrices which have vector [2 1 4 ] ^T   as an eigenvector

let A,B belong to this set ,then

A v = a v , B v = b v ,(A+B) v = av +bv = (a+b) v hence (A+ B) belong to this set too

iii)the set of all polynomials in P2 of hthe form a0 +a1 x+a2 x²  ,where a0 = a2

let   a0 +a1 x+a2 x^{2, ,,}b0 +b1 x+b2 x² belong to this set, a0 = a2 ,b0 = b2

now adding two we get ( a0 +a1 x+a2 x2, ) + (,b0 +b1 x+b2 x2 ) ,we get (a0+b0) + (a1+b1)x + (a2 +b2)x²  clearly this also belong to set as (a0+b0) = (a2 +b2)

hence answer is (G) ii) and iii) only