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

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

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


Proof

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

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

Then the sample matrix is:

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

Matrix $S$ is an invertible Vandermonde matrix.

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

$\blacksquare$