Linear Independence of Functions

If we have an $n^{\mathrm {th} }$ order linear homogenous differential equation ${\frac {d^{n}y}{dt^{n}}}+p_{1}(t){\frac {d^{n-1}y}{dt^{n-1}}}+...+p_{n-1}(t){\frac {dy}{dt}}+p_{n}(t)y=0$ where $p_{1}$, $p_{2}$, …, $p_{n}$ are continuous on an open interval $I$ and if $y=y_{1}(t)$, $y=y_{2}(t)$, …, $y=y_{n}(t)$ are solutions to this differential equation, then provided that $W(y_{1},y_{2},...,y_{n})\neq 0$ for at least one point $t\in I$, then $y_{1}$, $y_{2}$, …, $y_{n}$ form a fundamental set of solutions to this differential equation - that is, for constants $C_{1}$, $C_{2}$, …, $C_{n}$, then every solution to this differential equation can be written in the form:

${\begin{aligned}\quad y=C_{1}y_{1}(t)+C_{2}y_{2}(t)+...+C_{n}y_{n}(t)\end{aligned}}$

We will now look at the connection between the solutions $y_{1}$, $y_{2}$, …, $y_{n}$ forming a fundamental set of solutions and the linear independence/dependence of such solutions. We first define linear independence and linear dependence below.

Definition: The functions $f_{1}$, $f_{2}$, …, $f_{n}$ are said to be Linearly Independent on an interval $I$ if for constants $k_{1}$, $k_{2}$, …, $k_{n}$ we have that $k_{1}f_{1}(t)+k_{2}f_{2}(t)+...+k_{n}f_{n}(t)=0$ implies that $k_{1}=k_{2}=...=k_{n}=0$ for all $t\in I$. This set of functions is said to be Linearly Dependent if $k_{1}f_{1}(t)+k_{2}f_{2}(t)+...+k_{n}f_{n}(t)=0$ where $k_{1}$, $k_{2}$, …, $k_{n}$ are not all zero for all $t\in I$.

Perhaps the simplest linearly independent sets of functions is that set that contains $f_{1}(t)=1$, $f_{2}(t)=t$, and $f_{3}(t)=t^{2}$. Let $k_{1}$, $k_{2}$, and $k_{3}$ be constants and consider the following equation:

${\begin{aligned}\quad k_{1}f_{1}(t)+k_{2}f_{2}(t)+k_{3}f_{3}(t)=0\\\quad k_{1}+k_{2}t+k_{3}t^{2}=0\end{aligned}}$

It's not hard to see that equation above is satisfied if and only if the constants $k_{1}=k_{2}=k_{3}=0$.

For another example, consider the functions $f_{1}(t)=\sin t$ and $f_{2}(t)=\sin(t+\pi )$ defined on all of $\mathbb {R}$. This set of functions is not linearly independent. To show this, let $k_{1}$ and $k_{2}$ be constants and consider the following equation:

${\begin{aligned}\quad k_{1}f_{1}(t)+k_{2}f_{2}(t)=0\\\quad k_{1}\sin t+k_{2}\sin(t+\pi )=0\end{aligned}}$

Now choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = \pi} . Then we have that:

${\begin{aligned}\quad k_{1}\sin \pi +k_{2}\sin 2\pi =0\\\end{aligned}}$

But the above equation is true for any choice of constants $k_{1}$ and $k_{2}$ since $\sin \pi =\sin 2\pi =0$, and thus $f_{1}$ and $f_{2}$ do not form a linearly independent set on all of $\mathbb {R}$.

From the concept of linear independence/dependence, we obtain the following theorem on fundamental sets of solutions for $n^{\mathrm {th} }$ order linear homogenous differential equations.

Theorem 1: Let ${\frac {d^{n}y}{dt^{n}}}+p_{1}(t){\frac {d^{n-1}y}{dt^{n-1}}}+...+p_{n-1}(t){\frac {dy}{dt}}+p_{n}(t)y=0$ be an $n^{\mathrm {th} }$ order linear homogenous differential equation. If $y=y_{1}(t)$, $y=y_{2}(t)$, …, $y=y_{n}(t)$ are solutions to this differential equation then $y_{1}$, $y_{2}$, …, $y_{n}$ form a fundamental set of solutions to this differential equation on the open interval $I$ if and only if $y_{1}$, $y_{2}$, …, $y_{n}$ are linearly dependent on $I$.

• Proof: Consider the following Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0 \end{align}}

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow} Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_1(t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_2(t)} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_n(t)} form a fundamental set of solutions to this differential equation on the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . Then this implies that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0 \in I} we have that :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad W(y_1, y_2, ..., y_n) \big|_{t_0} \neq 0 \end{align}}

• Thus this implies that following system of equations have only trivial solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1 = k_2 = ... = k_n = 0} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad k_1y_1(t) + k_2y_2(t) + ... + k_ny_n(t) = 0 \\ \quad k_1y_1'(t) + k_2y_2'(t) + ... + k_ny_n'(t) = 0 \\ \quad \quad \quad \quad \quad \quad \vdots \quad \quad \quad \quad \quad \quad \\ \quad k_1y_1^{(n-2)}(t) + k_2y_2^{(n-2)}(t) + ... + k_ny_n^{(n-2)}(t) = 0 \\ \quad k_1y_1^{(n-1)}(t) + k_2y_2^{(n-1)}(t) + ... + k_ny_n^{(n-1)}(t) = 0 \end{align}}

• Thus the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1y_1(t) + k_2y_2(t) + ... + k_ny_n(t) = 0} implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1 = k_2 = ... = k_n = 0} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} are linearly independent on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} .

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow} We will prove the converse of Theorem 1 by contradiction. Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} are linearly independent on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} , and assume that instead Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} do NOT form a fundamental set of solutions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . Then for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0 \in I} , the Wronskian Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(y_1, y_2, ..., y_n) \right|_{t_0} = 0} . Thus the system of equations above does not have only the trivial solution. Let the constants $k_{1}^{*}$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_2^*} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_n^*} be a nontrivial solution to this system. Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t)} as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \phi(t) = k_1^* y_1(t) + k_2^* y_2(t) + ... + k_n^* y_n(t) \end{align}}

• Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \phi(t)} satisfies the initial conditions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t_0) = 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'(t_0) = 0} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^{(n-1)} (t_0) = 0} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t)} satisfies our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equation because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t)} is a linear combination of the solutions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} .

• Now note that the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 0} also satisfies the differential equation and the initial conditions. By the existence/uniqueness theorem for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equations, this implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t) = 0} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \in I} , so:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 0 = k_1^* y_1(t) + k_2^* y_2(t) + ... + k_n^* y_n(t) \end{align}}

• But Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} are linearly independent which implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1^* = k_2^* = ... = k_n^* = 0} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1^*} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_2^*} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_n^*} is a trivial solution to the system above, which is a contradiction. Therefore our assumption that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} do not form a fundamental set of solutions was false. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}