Sample Matrix Independence Test

Theorem
Let $V$ be a vector space of real-valued functions on a set $J$.

Let $f_1, \ldots, f_n$ be functions in $V$.

Let samples $x_1, \ldots, x_n$ from $J$ be given.

Define sample matrix


 * $\displaystyle S = \paren {\begin{smallmatrix}

f_1(x_1) & \cdots      & f_n(x_1) \\ \vdots       & \ddots & \vdots \\ f_1(x_n)  & \cdots     & f_n(x_n) \\ \end{smallmatrix} }$

Let $S$ be invertible.

Then $f_1, \ldots, f_n$ are linearly independent in $V$.

Proof
Let's apply the definition of linear independence.

Assume a linear combination of the functions is the zero function:


 * $\displaystyle \sum_{i \mathop = 1}^n c_i \, \map {f_i} x = 0$ for all $x$

Let $\vec c$ have components $c_1, \ldots, c_n$.

For $i = 1, \ldots, n$ replace $x = x_i$ in the linear combination equality.

There are $n$ linear homogeneous algebraic equations, written as:


 * $S \vec c = \vec 0$

Because $S$ is invertible, then $\vec c = \vec 0$.

The functions are linearly independent.

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


 * $\displaystyle y - y = 0$ Linearly Independent Solutions of y - y = 0

Details
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 = \paren {\begin{smallmatrix}

1 & 1 \\ e & 1/e \\ \end{smallmatrix} }$

Matrix $S$ is invertible.

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

Also see
Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent