TIFR-2018-Maths-A: 7

Question

True/False Question :

In the vector space $\left \{ f \mid f : \left [ 0,1 \right ] \rightarrow \mathbb{R}\right \}$ of real-valued functions on the closed interval  $\left [ 0,1 \right ]$, the set $S=\left \{ sin\left ( x \right ) , cos\left ( x \right ),tan\left ( x \right )\right \}$ is linearly independent.

Answer 1

$\text{For $c_1\sin x +c_2 \cos x + c_3 \tan x= 0$, }$

$\text{when $x=0, c_2=0$}$

$\text{when $x=\frac{\pi}{4}, \frac{c_1}{\sqrt{2}}+\frac{c_2}{\sqrt{2}}+c_3 =0 \Rightarrow c_1 + \sqrt{2}c_3 = 0 \;(\because c_2=0)$}$

$\text{when $x=\frac{\pi}{6}, \frac{c_1}{2}+\frac{\sqrt{3}c_2}{2}+\frac{c_3}{\sqrt{3}} =0 \Rightarrow \sqrt{3}c_1 + 2c_3 = 0 \;(\because c_2=0)$}$

$\text{Solving $c_1 + \sqrt{2}c_3 = 0$ and $\sqrt{3}c_1 + 2c_3 = 0$ gives $c_1=c_3=0$}$

$\text{Since, $c_1=c_2=c_3=0$, So, set $S$ is linearly independent.}$