Recall that subspaces of a vector space are non-empty sets which are closed under any linear combination in the set. We also talked about what is a subspace in R n
. Each of the following examples is not a subspace of the vector space R 3
and understanding when a set fails to be a subspace helps you understand the definition. For each, give an example of a linear combination of vectors in the set that fails to be in the set (and hence it is not a subspace).
(a) The set of vectors ⎝
⎛
x
y
z
⎠
⎞
such that xyz=0.
(b) The set of vectors ⎝
⎛
x
y
z
⎠
⎞
such that x≤y≤z.