Sample Matrix Independence Test
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Theorem
Let $V$ be a vector space of real or complex-valued functions on a set $J$.
Let $f_1, \ldots, f_n$ be functions in $V$.
Let $x_1, \ldots, x_n$ from $J$ be given.
Define the sample matrix:
- $S = \begin{bmatrix} \map {f_1} {x_1} & \cdots & \map {f_n} {x_1} \\ \vdots & \ddots & \vdots \\ \map {f_1} {x_n} & \cdots & \map {f_n} {x_n} \\ \end{bmatrix}$
Let $S$ be nonsingular.
Then $f_1, \ldots, f_n$ are linearly independent in $V$.
Proof
The definition of linear independence is applied.
Assume a linear combination of the functions $f_1, \ldots, f_n$ is the zero function:
| \(\text {(1)}: \quad\) | \(\ds \sum_{i \mathop = 1}^n c_i \map {f_i} x\) | \(=\) | \(\ds 0\) | for all $x$ |
Let $\vec c$ have components $c_1, \ldots, c_n$.
For $i = 1, \ldots, n$ replace $x = x_i$ in $(1)$.
There are $n$ linear homogeneous algebraic equations, written as:
- $S \vec c = \vec 0$
Because $S$ is nonsingular:
- $\vec c = \vec 0$
The functions are linearly independent.
$\blacksquare$
Examples
Example: Linearly Independent Solutions of $y - y = 0$
Prove independence of the solutions $e^x$, $e^{-x}$ to:
- $y' ' - y = 0$
Example: Linear Independence of Powers $1, x, \ldots, x^{n - 1}$
Let $V$ be the vector space of all continuous functions on $\R$.
Let $n$ be a strictly positive integer and define:
- $S = \set {1, x, \ldots, x^{n - 1} }$
Then $S$ is a linearly independent subset of $V$.