Representation of vectors by basis is Unique

[FourierX]

Homework Statement

Prove:

Representation of vectors by any basis is unique.

Homework Equations

The Attempt at a Solution

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

 

[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.

 

[HallsofIvy]

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.