Having difficulty with determining vector subspaces. I attached the file to bett

Question

Having difficulty with determining vector subspaces. I attached the file to better clarify what my questions are. Thank you for any help.

Below are some subsets of vector spaces. For each subset V, determine if V is a subspace (a vector space in its own right) or not.| V R3 defined by V = {[a b c] : a, b, c epsilon R and c = a + b}. V R3 defined by V = {[a 1 b] : a, b epsilon R}. V 0 defined by V = {alpha 1x3 + alpha 2x22 : alpha 1, alpha 2 epsilon R}. where 0 is the vector space of all continuous real-valued functions on R. V 0 defined by V = {alpha 1 x3 + alpha 2 x2 + 3 : alpha 1, alpha 2 epsilon R}.

Explanation / Answer

A subset of a vector space is just a set of elements from the vector space.

A subspace of a vector space is a subset that is a vector space itself under the same operations as the vector space.

So basically not every subset is a subspace, but every subspace is a subset.

If a subset has the following 3 properties than it is a subspace:
1) the zero vector is in the subset
2) if you add 2 elements of the subset they remain in the subset
3) if you multiply any element in the subset by a constant than this is in the subset.

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