# Representation of vectors by basis is Unique

1. Oct 29, 2008

### FourierX

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

Prove:

Representation of vectors by any basis is unique.

2. Relevant equations

3. The attempt at a solution

The minimal span set and the maximum linearly independent set gives a basis.

2. Oct 29, 2008

### gabbagabbahey

Try expanding a vector w into a basis set {w_i} using two different sets of coefficients i.e.

$\vec{w}=\sum_i a_i \hat{w}_i$ and $\vec{w}=\sum_i b_i \hat{w}_i$

Then just show that a_i=b_i for all i.

Last edited: Oct 29, 2008
3. Oct 29, 2008

### HallsofIvy

Staff Emeritus
gabbagabbahey's notation, think about the fact that
$\sum_i (a_i- b_i)\hat{w}_i= 0$ and use the fact that the vectors are independent.