Question

Which of the following sets are closed under addition? (i) The set of all vectors in R2 of the form (a, b) where b = a. (ii) The set of all 3 × 3 matrices that have the vector [-1  3  -3]T as an eigenvector. (iii) The set of all polynomials in P2 of the form a0 + a1 x + a2 x2 where a0 = a22.

Answer 1

0- lened inder Which of The llowing aeh re Addition S = a= nd (,)eS (4,4) 6S Tren (a,a) +I,6)-(ark, att at bat We con nee Sis desed Uncen Adelition Hences Matrices Tht hue The ALl nct of li) S= Tre 3X3 Ve tar eigen Veetir 3 letion A BeS ond eipen Vectr Sy a of Matsin A nd Bwith Rdeene t to Eigen Value di and

Thet is -1 1 -I d2 3 -3 Adding There Tuse - I (A+B) (Atde) 3 3 -3 eigen Vectar of A+R is am - 3 A B e S dlared Onder Addition pex) E Pb) S E axt ES umd

pir)+2lx)(4gt a) +l4t6) + (4 tha Ace Thet A Je Con The cofficint of The Comstont and Tom i Same. Hence boy+2() E S is clared Under Addition