# Sample Matrix Independence Test/Examples

## Examples of Sample Matrix Independence Test

### Example: Linearly Independent Solutions of $y'' - y = 0$

Prove independence of the solutions $e^x$, $e^{-x}$ to:

$\displaystyle y'' - y = 0$

### Example: Linear Independence of Powers $1,x,\ldots,x^{n-1}$

Let $V$ be the vector space of all continuous functions on $\R$.

Let $n \ge 1$ be an integer and define:

 $\ds S$ $=$ $\ds \set {1, x, \ldots, x^{n-1} }$

Prove that $S$ is a linearly independent subset of $V$.