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

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

Exponential Functions are Linearly Independent
Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers.
Show that exponential functions
\[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\]
are linearly independent over $\R$.
Hint.
Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\]
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Linear Independent Vectors and the Vector Space Spanned By Them
Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.
Let […]

Cosine and Sine Functions are Linearly Independent
Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$.
Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.
Proof.
Note that the zero vector in the vector space $C[-\pi, \pi]$ is […]

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