Linear Algebra

Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?

Ohio State University exam problems and solutions in mathematics

Problem 603

Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\] in $C[-2\pi, 2\pi]$.

Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly independent.

(The Ohio State University, Linear Algebra Midterm)

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Proof.

To determine whether $f(x)$ and $g(x)$ are linearly independent or not, consider the linear combination
\[c_1f(x)+c_2g(x)=0,\] equivalently
\[c_1\sin^2(x)+c_2 \cos^2(x)=0, \tag{*}\] where $c_1, c_2$ are scalars.

If the only scalars satisfying the above equality are $c_1=0, c_2=0$, then $f(x)$ and $g(x)$ are linearly independent, otherwise they are linearly dependent.

Note that this is an equality as functions.
That is, this equality must hold for any $x$ in the interval $[-2\pi, 2\pi]$.

Let $x=0$. Then as $\sin(0)=0$ and $\cos(0)=1$, we obtain $c_2=0$ from (*).
Next, let $x=\pi/2$. Then as $\sin(\pi/2)=1$ and $\cos(\pi/2)=0$, we obtain $c_1=0$ from (*).

Therefore, we must have $c_1=c_2=0$, and hence the functions $f(x)=\sin^2(x)$ and $g(x)=\cos^2(x)$ are linearly independent.

Comment.

This is one of the midterm 2 exam problems for Linear Algebra (Math 2568) in Autumn 2017.

Here is the most common mistake.
The linear combination $c_1\sin^2(x)+c_2 \cos^2(x)$ is a function defined over the interval $[-2\pi, 2\pi]$ and we are assuming it is the zero function.

So saying that “if $c_1=1, c_2=0$, then $c_1\sin^2(x)+c_2 \cos^2(x)$ is zero at $x=0$, hence $f(x)$ and $g(x)$ are linearly independent” is totally wrong.

What you are claiming here is that the function $\sin^2(x)$ is zero at $x=0$, hence it is the zero function.
This is clearly wrong as $\sin^2(x)$ is not the zero function.

List of Midterm 2 Problems for Linear Algebra (Math 2568) in Autumn 2017

Vector Space of 2 by 2 Traceless Matrices
Find an Orthonormal Basis of the Given Two Dimensional Vector Space
Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?←The current problem
Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix
Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$
Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors
Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less

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