Proof Linear Span Subsets: Proving L(S) is Smallest Subspace of V

[Caroline Fields]

This question asks me to prove a statement regarding linear spans. I have devised a proof but I am not sure if it is substantial enough to prove the statement.

1. Homework Statement
Let L(S) be the subspace spanned by a subset S of a linear space V. Prove that if S is a subset of T and T is a subset of V and if T is a subspace of V, then L(S) is a subspace of T. (This property is described by saying that L(S) is the smallest subspace of V which contains S).

Homework Equations

In part (a) of the question I proved that for S={u₁, ..., u_n}, S is always a subset of L(S).

The Attempt at a Solution

From the definition of the span of S, we know that L(S) is the smallest subspace of V that contains S. Therefore, for any T that is a subset of V such that S is a subset of T, it is clear from the definition of span that L(S) is a subset of T.
Is this proof substantial enough?

 

[PeroK]

This question asks me to prove a statement regarding linear spans. I have devised a proof but I am not sure if it is substantial enough to prove the statement.

1. Homework Statement
Let L(S) be the subspace spanned by a subset S of a linear space V. Prove that if S is a subset of T and T is a subset of V and if T is a subspace of V, then L(S) is a subspace of T. (This property is described by saying that L(S) is the smallest subspace of V which contains S).

Homework Equations

In part (a) of the question I proved that for S={u₁, ..., u_n}, S is always a subset of L(S).

The Attempt at a Solution

From the definition of the span of S, we know that L(S) is the smallest subspace of V that contains S. Therefore, for any T that is a subset of V such that S is a subset of T, it is clear from the definition of span that L(S) is a subset of T.
Is this proof substantial enough?

No. Because you don't use the fact that T is a subspace.
You can use the fact that L(S) is the smallest subspace containing S or consider L(T).