Let 5 denote the set of all vectors x y z in R3 whose compon

Let 5 denote the set of all vectors (x, y, z) in R^3 whose components satisfy the condition given. Determine whether S is a subspace of R^3. If S is a subspace, compute dim x = y or x = z x^2 - y^2 = 0 x + y + z = 0 and x - y - z = 0

Solution

The Subspace Test To test whether or not S is a subspace of some Vector Space R n you must check two things:

1. if s1 and s2 are vectors in S, their sum must also be in S

2. if s is a vector in S and k is a scalar, ks must also be in S

Following all the three conditons

there is only one and only one possibility which is that all the elements of this subspace becomes zero

and being zero it will satisfy everything thus S will be a subspace and dimension will be 3.