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 $n^{\mathrm {th} }$ order linear homogenous differential equation:

${\begin{aligned}\quad {\frac {d^{n}y}{dt^{n}}}+p_{1}(t){\frac {d^{n-1}y}{dt^{n-1}}}+...+p_{n-1}(t){\frac {dy}{dt}}+p_{n}(t)y=0\end{aligned}}$

• $\Rightarrow$ Suppose that $y=y_{1}(t)$, $y=y_{2}(t)$, …, $y=y_{n}(t)$ form a fundamental set of solutions to this differential equation on the open interval $I$. Then this implies that for all $t_{0}\in I$ we have that :

${\begin{aligned}\quad W(y_{1},y_{2},...,y_{n}){\bigg |}_{t_{0}}\neq 0\end{aligned}}$

• Thus this implies that following system of equations have only trivial solution $k_{1}=k_{2}=...=k_{n}=0$:

${\begin{aligned}\quad k_{1}y_{1}(t)+k_{2}y_{2}(t)+...+k_{n}y_{n}(t)=0\\\quad k_{1}y_{1}'(t)+k_{2}y_{2}'(t)+...+k_{n}y_{n}'(t)=0\\\quad \quad \quad \quad \quad \quad \vdots \quad \quad \quad \quad \quad \quad \\\quad k_{1}y_{1}^{(n-2)}(t)+k_{2}y_{2}^{(n-2)}(t)+...+k_{n}y_{n}^{(n-2)}(t)=0\\\quad k_{1}y_{1}^{(n-1)}(t)+k_{2}y_{2}^{(n-1)}(t)+...+k_{n}y_{n}^{(n-1)}(t)=0\end{aligned}}$

• Thus the equation $k_{1}y_{1}(t)+k_{2}y_{2}(t)+...+k_{n}y_{n}(t)=0$ implies that $k_{1}=k_{2}=...=k_{n}=0$. Thus $y_{1}$, $y_{2}$, …, 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) \bigg|_{t_0} = 0} . Thus the system of equations above does not have only the trivial solution. Let the constants 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^*} , …, $k_{n}^{*}$ 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}

Licensing

Content obtained and/or adapted from:

• Linear Independence Dependence of a Set of Functions, http://mathonline.wikidot.com/ under a CC BY-SA license