Use the definitions for span and subspace to explain why the span of a set of ve

Question

Use the definitions for span and subspace to explain why the span of a set of vectors from V generates a subspace of V.

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Definitions:

Span: Given a vector space V over a field K (scalars), the span of a set S (not necessarily finite) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W. The span of S may also be defined as the set of all linear combinations of the elements of S.

Subspace: Let K be a field, and let V be a vector space over K. As usual, we call elements of V vectors and call elements of K scalars. Suppose that W is a subset of V. If W is a vector space itself, with the same vector space operations as V has, then it is a subspace of V.

Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following three conditions:

Explanation / Answer

Because the span of V is the set of all linear combinations of V, additivity and scalar multiplication will always be closed.

For example, Let w1=x1v1+...+xnvn and w2=y1v'1+...+ymv'm be two linear combinations of elements from V.

Then w1+w2 is equal to x1v1+...+xnvn+y1v'1+...+ymv'm which, like w1 and w2, is a linear combination of elements from V. This shows us that the sum of any two elements of W is itself an element of W, and that W is closed under addition.

To prove closure of scalar multiplication, take another arbitrary linear combination w=x1v1+...xnvn of elements in V and an arbitrary scalar k. Then the product kw is equal to kx1v1+...+kxnvn, which, again, is a a linear combination of elements of V. Therefore W is closed under scalar multiplication, and is therefore a subspace a V.

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