1. Vector Spaces & Subspaces Kristi Schmit

2. Definitions A subset W of vector space V is called a subspace of V iff a.The zero vector of V is in W. b.W is closed under vector addition, for each u and v in W, the sum u + v is in W. c.W is closed under multiplication by scalars, for each u in W and each scalar c, the vector c u is in W. Any subspace W of vector space V is a vector space.

3. Example 1 Let W be the set of all vectors of the form shown, where a and b represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is not a vector space.

4. Response The zero vector of V is not in W because of the -1 in the subset. Therefore the subset fails the first property of a subspace. Thus, W is not a subspace of V and therefore is not a vector space.

5. Definitions If v 1,…., v p are in an n-dimensional vector space over the real numbers, R n, then the set of all linear combinations of v 1,…., v p is denoted by Span{ v 1,…., v p } and is called the subset of R n spanned by v 1,…., v p. That is, Span{ v 1,…., v p } is the collection of all vectors that can be written in the form c 1 v 1 + c 2 v 2 + …+ c p v p with c 1,…, c p scalars.

6. Example 2 Let W be the set of all vectors of the form shown, where a and b represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is not a vector space.

7. Response the set of all linear combinations of v 1,…., v p