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

Linear algebra problems and solutions

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 $[-1, 1]$.

 
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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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