SOLUTION: Determine whether the following sets are subspaces of R3: W1 = {f(x; y; z)R3 : x - 4y - z = 0}; W2 = {f(x; y; z)R3 : x + 2y - 3z = 1} and W3 = {f(x; y; z)R3 : 5x2 + 3y2 + 6z2 = 0}

Question 1094592: Determine whether the following sets are subspaces of R3: W1 = {f(x; y; z)R3 : x - 4y - z = 0};
W2 = {f(x; y; z)R3 : x + 2y - 3z = 1} and W3 = {f(x; y; z)R3 : 5x2 + 3y2 + 6z2 = 0}:
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
W1, x -4y -z = 0
This is a subspace because it's the set of solutions of a homogeneous linear system
:
W2, x +2y -3z = 1
This is not a subspace because there is no zero vector
:
W3, 5x^2 +3y^2 +6z^2 = 0
This is not a subspace, because it is not closed under addition.
For instance, although (1/√5, 1/√3, 0) and (1/√5, -1/√3, 0) are in W3,
their sum (2/√5, 0, 0) is not in W3.
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