Linear Independent Continuous Functions

Linear Algebra Problems and Solutions

Problem 158

Let $C[3, 10]$ be the vector space consisting of all continuous functions defined on the interval $[3, 10]$. Consider the set
\[S=\{ \sqrt{x}, x^2 \}\] in $C[3,10]$.

Show that the set $S$ is linearly independent in $C[3,10]$.

 
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Proof.

Note that the zero vector in $C[3,10]$ is the zero function $\theta(x)=0$.
Let us consider a linear combination
\[a_1\sqrt{x}+a_2x^2=\theta(x)=0.\]

We want to show that $a_1=a_2=0$.
Since this equality holds for any value of $x$ between $3$ and $10$, letting $x=4$ and $x=9$ yields the system of linear equations
\begin{align*}
2a_1+4a_2=0\\
3a_1+81a_2=0.
\end{align*}
Solving this system, we obtain $a_1=a_2=0$, and hence the set $S$ is linearly independent.

Comment.

We could have taken any two values between $3$ and $10$ for $x$ instead of $4$ and $9$, but the choice $x=4$ and $x=9$ made solving the system easier.
Also, we took two values for $x$ because we had two unknowns $a_1$, $a_2$, and thus we needed two equations to determine these unknown.

For a similar problem, show that the set $\{\sqrt{x}, x^2, x\}$ in the vector space $C[1, 10]$ is linearly independent.
In this case, there will be three unknowns that you want to show to be zero.
So you need to take three values for $x$.
A natural choice will be $x=1, 4, 9$.


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