Question

Question No:1 (a) The set of all pairs of real numbers of the form (1,x) with the Operations (1,y)+(1,y')=(1,y+y') and k(1,y)=(1,ky) is vector space or not? (b) Show that the set of all matrices of the form [a 1; 1 b] with addition defined by [a 1; 1 b]+[c 1; 1 d] = [a+c 1; 1 b+d] and scalar multiplication defined by k [a 1; 1 b] = [ka 1; 1 kb] is a vector space. What is the zero vector in this space? Question No:2 Assume that V1,V2, and V3 are vectors in R^3 that have their initial point Origin. In each part determine whether the three vectors lie in a plane Or in a line. (i) v_1 = (2,-2,0) , v_2 = (6,1,4) , v_3 = (2,0,-4) (ii) v_1 = (-6,7,2) , v_2 = (3,2,4) , v_3 = (4,-1,2) Question No:3 A line L through the origin in R^3 can be represented by the Parametric equations of the form x = at , y = bt , z = ct. Use These equations to show that L is a subspace of R^3.

          Question No:1 (a) The set of all pairs of real numbers of the form (1,x) with the
Operations (1,y)+(1,y')=(1,y+y') and k(1,y)=(1,ky) is vector space or not?
(b) Show that the set of all matrices of the form [a 1; 1 b] with
addition defined by [a 1; 1 b]+[c 1; 1 d] = [a+c 1; 1 b+d]
and scalar multiplication defined by k [a 1; 1 b] = [ka 1; 1 kb] is a vector
space. What is the zero vector in this space?
Question No:2 Assume that V1,V2, and V3 are vectors in R^3 that have their initial point
Origin. In each part determine whether the three vectors lie in a plane
Or in a line.
(i) v_1 = (2,-2,0) , v_2 = (6,1,4) , v_3 = (2,0,-4)
(ii) v_1 = (-6,7,2) , v_2 = (3,2,4) , v_3 = (4,-1,2)
Question No:3 A line L through the origin in R^3 can be represented by the
Parametric equations of the form x = at , y = bt , z = ct. Use
These equations to show that L is a subspace of R^3.

Question No:1 (a) The set of all pairs of real numbers of the form (1,x) with the
Operations (1,y)+(1,y')=(1,y+y') and k(1,y)=(1,ky) is vector space or not?
(b) Show that the set of all matrices of the form [a 1; 1 b] with
addition defined by [a 1; 1 b]+[c 1; 1 d] = [a+c 1; 1 b+d]
and scalar multiplication defined by k [a 1; 1 b] = [ka 1; 1 kb] is a vector
space. What is the zero vector in this space?
Question No:2 Assume that V1,V2, and V3 are vectors in R^3 that have their initial point
Origin. In each part determine whether the three vectors lie in a plane
Or in a line.
(i) v1 = (2,-2,0) , v2 = (6,1,4) , v3 = (2,0,-4)
(ii) v1 = (-6,7,2) , v2 = (3,2,4) , v3 = (4,-1,2)
Question No:3 A line L through the origin in R^3 can be represented by the
Parametric equations of the form x = at , y = bt , z = ct. Use
These equations to show that L is a subspace of R^3.

Added by Melissa S.

Question

Please give Ace some feedback