# Linear Algebra Spanning Sets

1. Mar 30, 2010

### gutnedawg

1. The problem statement, all variables and given/known data

Let V be a vector space, let p ≤ m, and let b1, . . . , bm be vectors in V such that
A = {b1, . . . , bp} is a linearly independent set, while C = {b1, . . . , bm} is a spanning set
for V . Prove that there exists a basis B for V such that A ⊆ B ⊆ C.

2. Relevant equations

3. The attempt at a solution

I'm going on the fact that it does not mention C is linearly independent, thus by the spanning set theorem there exists a linearly independent set of vectors {bi,...,bk} which spans V. Thus, this set {bi,...,bk} is a basis for V.

This means that the basis must at least be equal to A since B cannot be a basis for V if there is another linearly independent vecotr bp. Meaning:

$$A \subseteq B$$

Also since B is a spanning set of V and is comprised of at least {b1,...,bp} it must be a subset of C since C also spans V and includes A.

Thus

$$A \subseteq B \subseteq C$$

Last edited: Mar 30, 2010