# Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent

## Problem 549

By calculating the Wronskian, determine whether the set of exponential functions
$\{e^x, e^{2x}, e^{3x}\}$ is linearly independent on the interval $[-1, 1]$.

## Solution.

The Wronskian for the set $\{e^x, e^{2x}, e^{3x}\}$ is given by
$W(x):=\begin{vmatrix} e^x & e^{2x} & e^{3x} \\ e^x &2e^{2x} &3e^{3x} \\ e^x & 4e^{2x} & 9e^{3x} \end{vmatrix}.$

We compute
\begin{align*}
W(x)&=e^xe^{2x} e^{3x}\begin{vmatrix}
1 & 1 & 1 \\
1 &2 &3 \\
1 & 4 & 9
\end{vmatrix}\6pt] &=2e^{6x}. \end{align*} Since the Wronskian W(x)=2e^{6x} is never zero, we conclude that the set \{e^x, e^{2x}, e^{3x}\} is linearly independent. (Note that in general, we just need to show that W(x_0)\neq 0 for some point x_0 \in [0, 1]. For example, since we have W(0)=2\neq 0, we can conclude that \{e^x, e^{2x}, e^{3x}\} is linearly independent.) ## Related Question. Try the following generalized version. Let c_1, c_2,\dots, c_n be mutually distinct real numbers. Show that functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx} are linearly independent over $\R$.

The solution is given by Exponential Functions are Linearly Independent.

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