Determine Whether Trigonometry Functions $\sin^2(x), \cos^2(x), 1$ are Linearly Independent or Dependent

Ohio State University exam problems and solutions in mathematics

Problem 365

Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$.
Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent.

(The Ohio State University, Linear Algebra Midterm Exam Problem)
 
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Solution.

We claim that the set is linearly dependent. To show the claim, we need to find nontrivial scalars $c_1, c_2, c_3$ such that
\[c_1 f(x)+c_2 g(x)+ c_3 h(x)=0.\]

From trigonometry, we know the identity
\[\sin^2(x)+\cos^2(x)=1.\] This implies that we have
\[\sin^2(x)+\cos^2(x)-1=0.\]

So we can choose $c_1=1, c_2=1, c_3=-1$, and thus the set is linearly dependent.

Linear Algebra Midterm Exam 2 Problems and Solutions


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5 Responses

  1. 04/07/2017

    […] Problem 2 and its solution: Determine whether trigonometry functions $sin^2(x), cos^2(x), 1$ are linearly independent or dependent […]

  2. 04/07/2017

    […] Problem 2 and its solution: Determine whether trigonometry functions $sin^2(x), cos^2(x), 1$ are linearly independent or dependent […]

  3. 04/07/2017

    […] Problem 2 and its solution: Determine whether trigonometry functions $sin^2(x), cos^2(x), 1$ are linearly independent or dependent […]

  4. 08/11/2017

    […] Problem 2 and its solution: Determine whether trigonometry functions $sin^2(x), cos^2(x), 1$ are linearly independent or dependent […]

  5. 10/18/2017

    […] Problem 2 and its solution: Determine whether trigonometry functions $sin^2(x), cos^2(x), 1$ are linearly independent or dependent […]

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