Since vector spaces can exist regardless of a matrix I wanted to see if we could do all the proofs for the general vector-spaces without using theory for matrices. Then it was only two proofs of the general vector spaces in my book that uses the theory of matrices. One was the rank-nullity theorem and the other one I will write below. On wikipedia there is a proof for the rank-nullity theorem where they don't consider a matrix, so that is ok, but I have the theorem below also:

> 
> If a vector space V has a basis $\beta=\{b\_1,...,b\_n\}$, then any set
>  in V containing more than n vectors must be linerly dependent.
> 
> 
> 

The sad thing is that in this proof they used theory about matrices from an earlier chapter. However let's say you didn't even know about matrices. Now is it possible to prove this without using matrix-theory?

**NOTE:**
Here are some things we may **not use** in proving this, because these properties are only proved after the theorem above is proved.

1. If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors.
2. If V has dimension p, then any p linerly independent vectors is a basis for V. And if p elements span V, then they are automatically a basis.

Basically what we **can use** are these things developed before the theorem:

1.The definition of the general vector space with 10 properties.

2.Definition of a subspace.

3. The definition of linear independence.
4. The definiton of a basis, which says that the vectors in the basis are linearly 
independent and every vector in the vector space is a linear combination of them.
5. Then unique representation of each vector in the vector space in terms of its scalar infront of each basis vector.

Is it then possible to prove the theorem above by this?, or would the theory of general vector spaces not be able to exist without matrices?