# Show that the linear span of a subset is a subset of the linear span of the superset.

Let V be a vector space over the universal set and S and T be two non-empty finite sets with S $$\subset$$ T. Show that $$$$ $$\subset$$ $$$$ where denotes the linear span of A.

I did something like this: since S is a subset of T, every element in S can be expressed using the basis of T, but I don't know how we can eliminate some elements from the basis of T to make the basis of S.

Also, it is not possible that their linear spans are the same?

• Write down the definition of a linear span, and the definition of inclusion, then apply the assumption to these definitions. May 28, 2020 at 10:28
• @AsafKaragila I'm not able to understand how to go about it. May 28, 2020 at 10:31
• See some general advice: karagila.org/2015/how-to-solve-your-problems May 28, 2020 at 10:31

Hint-If $$S$$ is a subset of $$T$$ , every vector in $$S$$ is a vector in $$T$$. Now $$Span(T)$$ is , by definition the smallest vector space containing $$T$$, and $$Span(S)$$ is the smallest vector space containing $$S$$ .