Can a Basis B Exist in V Such That A is Contained in B and B in C?

[gutnedawg]

Homework Statement

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.

Homework Equations

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

$$A \subseteq B \subseteq C$$

 

[mighty2000]

Therefore, there exists a basis B for V such that A ⊆ B ⊆ C.



I would like to add that this proof relies on the assumption that the vectors in C are not linearly dependent. If there are any linearly dependent vectors in C, then the statement may not hold true and further investigation would be needed to prove the existence of a basis B for V with the given properties. It is also worth mentioning that this proof only works for finite-dimensional vector spaces. For infinite-dimensional vector spaces, a different approach may be needed to prove the given statement.