Can someone answer this question: If all linear combinations of the elements of subset W which is part of V are in W as well, then W is a subset of V. (T/F)

Question

anonymous · Answer

Such a poorly worded question!

mathmate · Answer

Since your question said that W is a subset of V, then automatically W is a subset of V (true).

I suspect, however, you have paraphrased the question which actually refers to subspace, something along the following lines:

"If all linear combinations of the elements of subset W which is part of V are in W as well, then W is a SUBSPACE of V."

For a subset W of V to be a subspace, it has to satisfy two conditions:
1. The zero vector belongs to W
2. For every u, v in W, k is in Field of the space V, u+v is in W and ku is in W.

Can you infer from the question if the two conditions are satisfied?  If yes, the answer is true.