Sample Matrix Independence Test/Examples/Linearly Independent Solutions of y'' - y = 0

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Example of Sample Matrix Independence Test: Linearly Independent Solutions of $y'' - y = 0$

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

$\displaystyle y'' - y = 0$


Proof

Choose samples $x_1 = 0$, $x_2 = 1$ from set $J = \R$.

Define $\map {f_1} x = e^x$, $\map {f_2} x = e^{-x}$.

Then the sample matrix is:

$\displaystyle S = \begin{bmatrix} 1 & 1 \\ e & 1/e \\ \end{bmatrix}$

Matrix $S$ is invertible.

Then $\map {f_1} x = e^x$, $\map {f_2} x = e^{-x}$ are linearly independent.

$\blacksquare$


Also see

Linearly Independent Solutions of y'' - y = 0 by the Wronskian test