Exponential Functions are Linearly Independent

Problem 73

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 $\R$.

Contents

Hint.

1. Consider a linear combination $a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.$
2. Differentiate this equality $n-1$ times and you will get $n$ equations.
3. Write a matrix equation for the system. You will see the Vandermonde matrix.

Proof.

Suppose that we have a linear combination of these functions that is zero.
Namely, suppose we have
$a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0$ for some real numbers $a_1, a_2, \dots, a_n$.

We want to show that the coefficients $a_1, a_2, \dots, a_n$ are all zero.

By differentiating the equation, we obtain
$a_1c_1e^{c_1x}+a_2c_2e^{c_2x}+\cdots +a_n c_n e^{c_n x}=0.$ Differentiating repeatedly we further obtain the equalities
\begin{align*}
& a_1c_1^2e^{c_1x}+a_2c_2^2e^{c_2x}+\cdots +a_n c_n^2 e^{c_n x}=0\\
& a_1c_1^3e^{c_1x}+a_2c_2^3e^{c_2x}+\cdots +a_n c_n^3 e^{c_n x}=0\\
& \dots \\
& a_1c_1^{n-1}e^{c_1x}+a_2c_2^{n-1}e^{c_2x}+\cdots +a_n c_n^{n-1}
e^{c_n x}=0\\
\end{align*}

We rewrite these $n$ equations into the following matrix equation.
$\begin{bmatrix} 1 & 1 & \dots &1 \\ c_1 & c_2 & \dots & c_n \\[3pt] c_1^2 & c_2^2 & \dots & c_n^2 \\[3pt] \vdots & \vdots & \vdots & \vdots \\[3pt] c_1^{n-1} & c_2^{n-1} & \dots & c_n^{n-1} \end{bmatrix} \begin{bmatrix} a_1 e^{c_1x} \\ a_2 e^{c_2x} \\ \vdots \\ a_n e^{c^nx} \end{bmatrix}=\begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} \tag{*}$

The determinant of the left matrix is
$\det \begin{bmatrix} 1 & 1 & \dots &1 \\ c_1 & c_2 & \dots & c_n \\[3pt] c_1^2 & c_2^2 & \dots & c_n^2 \\[3pt] \vdots & \vdots & \vdots & \vdots \\[3pt] c_1^{n-1} & c_2^{n-1} & \dots & c_n^{n-1} \end{bmatrix}=\prod_{i<j}(c_j-c_i)$ by the Vandermonde determinant.

Since by assumption $c_1,\dots, c_n$ are distinct, the determinant is not zero.
Therefore by multiplying equality (*) by the inverse on the left, we obtain
$\begin{bmatrix} a_1 e^{c_1x} \\ a_2 e^{c_2x} \\ \vdots \\ a_n e^{c^nx} \end{bmatrix}=\begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix}.$ Since the functions $e^{c_i x}$ are always positive, we must have $a_1=a_2=\cdots=a_n=0$ as required.
Therefore the functions $e^{c_1 x}, \dots, e^{c_n x}$ are linearly independent.

Comment.

The determinant that we considered above is called the Wronskian for the set of functions $\{e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\}$.

Related Question.

The following problems are more concrete versions of the current problem.

Problem.
Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let
$V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}$ be a subset in $C[-1, 1]$.

(a) Prove that $V$ is a subspace of $C[-1, 1]$.
(b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.
(c) Prove that
$B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}$ is a basis for $V$.

See the post ↴
Exponential Functions Form a Basis of a Vector Space
for the solution.

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 solutions is given in the post ↴
Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent

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1. 08/28/2017

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