Question

1.prove that the following three functions are linearly dependent

Answer 1

Proof:

Given any functions $f_{1}(x),f_{2}(x),f_{3}(x)$ they are linearly dependent if we can find values of $c_{1},c_{2},c_{3}$ such that

$c_{1}f_{1}(x)+c_2f_{2}(x)+c_{3}f_{3}(x)=0$

Using the given functions in the above equation we get

$c_{1}f_{1}(x)+c_2f_{2}(x)+c_{3}f_{3}(x)=0\\\\c_{1}x^{2}+c_{2}(1-x^{2})+c_{3}(2+x^{2})=0\\\\\Rightarrow (c_{1}-c_{2}+c_{3})x^{2}+c_{1}+c_{2}+2c_{3}=0$

This will be satisfied if and only if

$c_1-c_2+c_3=0,c_1+c_2+2c_3=0$

Solving the equations we get

$c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0$

Since we have 3 variables and 2 equations thus we will get many solutions  

one being if we put $c_3=1$ we get

$c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0\\\\c_1=\frac{-3}{2}\\\\c_2=\frac{-1}{2}$

Thus we have $c_1=\frac{-3}{2},c_2=\frac{-1}{2},c_3=1$ as one solution. Hence the given functions are linearly dependent.