Question 836555: which of the following sets are subspaces of R^n
a){(x,1,-6) | x arbitrary number}
b){x,y,z | -7x^2-4y^2-8z^2 <= 1}
c){(0,6y,0 | y arbitrary number}
d){(4x,y,z | y-z=0}
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Remember, a set is a subspace of  if it is closed under addition and scalar multiplication.
a) is not a subspace because (x1, 1, -6) + (x2, 1, -6) = (x1+x2, 2, -12) which is not in the original set.
b) is not a subspace since we can scalar multiply an element (x,y,z) by a large enough constant so that -7x^2 - 4y^2 - 8z^2 > 1.
c) is a subspace since (0, 6y_1, 0) + (0, 6y_2, 0) = (0, 6(y_1 + y_2), 0), and k(0, 6y, 0) = (0, 6ky, 0).
d) is a subspace since the condition implies y = z; you can check that the closure conditions hold.
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