Respuesta :
Proof:
Given any functions [tex]f_{1}(x),f_{2}(x),f_{3}(x)[/tex] they are linearly dependent if we can find values of [tex]c_{1},c_{2},c_{3}[/tex] such that
[tex]c_{1}f_{1}(x)+c_2f_{2}(x)+c_{3}f_{3}(x)=0[/tex]
Using the given functions in the above equation we get
[tex]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[/tex]
This will be satisfied if and only if
[tex]c_1-c_2+c_3=0,c_1+c_2+2c_3=0[/tex]
Solving the equations we get
[tex]c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0[/tex]
Since we have 3 variables and 2 equations thus we will get many solutions
one being if we put [tex]c_3=1[/tex] we get
[tex]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}[/tex]
Thus we have [tex]c_1=\frac{-3}{2},c_2=\frac{-1}{2},c_3=1[/tex] as one solution. Hence the given functions are linearly dependent.
