Determine if the following subsets of R3 are subspaces Justi

Determine if the following subsets of R^3 are subspaces. Justify your answers. (a) {(x_1,x_2,x_3)^T|x_1 + x_2 =0 and x_2 + x_3 = 0} (b) {(x_1,x_2,x_3)^T|x_1 + x_2 = 0 and x_2 + x_3 = 0}

Solution

You can solve the both the euation simultaneously as

x₁+x₂ =0 & x₂ + x₃ =0 gives x₁=x₃ & x₂=-x₁
So from here we can get

{x_1,x₂,x₃} ={1,-1,1} or {k,-k,k} where k belongs to R.