Determine if the given functions are linearly dependent or independent in (0, infinity). cos (ln...

Question:

Determine if the given functions are linearly dependent or independent in {eq}\displaystyle (0,\ \infty) {/eq}.

{eq}\displaystyle \cos (\ln x),\ \sin (\ln x) {/eq}.

Wronskian:

The Wronskian of a system of functions allows to analyze if the said system is linearly independent:

If the Wronskian is nonzero, we will say that the system is linearly independent.

Answer and Explanation:

Given the functions: {eq}\displaystyle \cos (\ln x),\ \sin (\ln x) {/eq}, calculating the wronskian:

{eq}W\left( {\cos (\ln x),\;\sin (\ln x)} \right) = \left| {\begin{array}{*{20}{c}} {\cos (\ln x)}&{\sin (\ln x)}\\ { - \sin (\ln x)\frac{1}{x}}&{\cos (\ln x)\frac{1}{x}} \end{array}} \right|\\ = {\cos ^2}(\ln x)\frac{1}{x} + {\sin ^2}(\ln x)\frac{1}{x}\\ = \left( {{{\cos }^2}(\ln x) + {{\sin }^2}(\ln x)} \right)\frac{1}{x}\\ = \frac{1}{x} \ne 0 {/eq}

As the result is not equal to zero, we can say the function are linearly independent.


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Linearization of Functions

from Math 104: Calculus

Chapter 10 / Lesson 1
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