# Proof: Max number of Linearly Independent Vectors

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1. Sep 13, 2015

### #-Riley-#

1. The problem statement, all variables and given/known data
Prove that a set of linearly independent vectors in Rn can have maximum n elements.

So how would you prove that the maximum number of independent vectors in Rn is n?

I can understand why in my head but not sure how to give a mathematical proof. I understand it in terms of the number of independent vectors being equal to the rank of the matrix they create and obviously a matrix of dimension n can only have max n pivots. But I don't think that's really sufficient for a proof.

Last edited: Sep 13, 2015
2. Sep 13, 2015

### Staff: Mentor

If you write it down properly, that should be fine.
The main point: a n x m matrix (for your m vectors) cannot have rank larger than n.

3. Sep 13, 2015

### Ray Vickson

Yes, it is (with a bit of tweaking). If vectors $v_1, v_2, \ldots, v_n, v_{n+1} \in R^n$ are linearly independent, we should not be able to find $(c_1, c_2, \ldots, c_{n+1}) \neq(0,0, \ldots, 0)$ giving $c_1 v_2 + c_2 v_2 + \cdots + c_n v_n + c_{n+1} v_{n+1} = \vec{0}$. Assuming, instead, that you CAN find such $c_i$, you should be able to get a contradiction.

Last edited by a moderator: Sep 13, 2015