In the following check to see if the set S is a vector subsp

In the following check to see if the set S is a vector subspace of the corresponding R^n. If it is not, explain why not. If it is, then find a basis and the dimension. S = {[x_1 x_2 x_3 x_4], x_1 + x_2 + x_3 = 0} R^4

Solution

yes S is a vector subspace of the corresponding R^n. As S spans R and S is linearly independent.

S = {x1, x2, x3 , x4}

{x1,x2,x3,x4} is a basis and dimension = 4 , no. of vectors in basis