Determine which of the following are subspaces Justify your

Determine which of the following are subspaces. (Justify your answer) The set of vectors (a, b, c) such that c = a + b - 2 The set of all vectors of the form (a, b, 0) W = {(x_1, x_2, x_1x_2): x_1 and X_2 are real numbers}.

Solution

A subset S is a subspace if all of the following are true:

1) If it contains 0.
2) When any vector in S is closed under addition. 3) If it\'s closed under multiplication.

Solution (a): 0 is not in (a,b,c), since a+b-c=2 or any other combination of values for a , b and c does not produce the zero vector. So property fails to hold and therefore (a,b,c) is not a subspace.

Solution(b): (a,b,0) is a subspace since all the conditions of vector space is met with.

Solotion(c) : (x_1,x₂,X₁X₂) is not a subspace since it is neither closed under addition nor closed under multiplication

Eg: consider S=(1,2,1x2)

S+S=(1,2,2)+(1,2,2)=(2,4,4) does not satisfy (x_1,x₂,x₁x₂)

3S=3x(1,2,2)=(3,6,6) does not satisfy (x_1,x₂,x₁x₂)