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

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)

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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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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