Sample Matrix Independence Test/Examples/Linear Independence of Powers

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Example of Sample Matrix Independence Test: Linear Independence of Powers

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

Let $n$ be a strictly positive integer and define:

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

Then $S$ is a linearly independent subset of $V$.

Proof

Choose samples $x_j = j$, $j = 1, \ldots, n$ from $\R$.

Define:

$\map {f_j} x = x^{j - 1}$ for $1 \le j \le n$.

Then the sample matrix is:

$S = \begin{bmatrix} 1 & 1 & \cdots & 1\\ 1 & 2 & \cdots & 2^{n - 1} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & n & \cdots & n^{n - 1} \\ \end{bmatrix}$

Matrix $S$ is an invertible Vandermonde matrix.

Then functions $f_1, \ldots, f_n$ are linearly independent.

$\blacksquare$