Check whether f1 x 1 f2 x x and f3 x ex are a linear inde

Check whether f_1 (x) = 1, f_2 (x) = x, and f_3 (x) = e^x are a linear independent set of functions.

Solution

if 3 functions are linearly independent then the only way for a f(x) + b g(x) + c h(x) =0 will be if a=b=c=0;
Hence a (1) + b(x) + c(e^x) =0 ;
Differentiate with respect to x we get :
b+ ce^x =0 ; Thus when e^x= -b/c or x= ln (-b/c) the equation becomes zero;
Hence it need not be necessary taht a=b=c=0 for the case;
Thus the given 3 funcions are not linearly independent or are linearly dependent