3.3.7

Let V be the vector space of functions defined on the interval [0,1] over the field $$\mathbb{R}$$, and let $$f_1,f_2,f_3 $$ be the elements in V given by $$f_1(x) = x^3, \ f_2(x) = sin (x), \  f_3(x) = cos(x)$$. We claim that these three vectors are linearly independent. Suppose there exist real numbers $$c_1,c_2,c_3$$ such that $$c_1f_1 + c_2f_2 + c_3f_3 = 0_V$$, where $$0_V$$ is the zero function. Then the relation $$c_1f_1(x) + c_2f_2(x) + c_3f_3(x) = c_1x^3 + c_2sin(x) + c_3cos(x) = 0$$ holds for every x in the interval [0,1]. In particular, it must hold at x = 0, so that $$c_3 = 0$$, reducing the linear relation to $$c_1x^3 + c_2sin(x) = 0$$. Differentiating with respect to x, we have $$3c_1x^2 + c_2cos(x) = 0$$ for every x in [0,1]. Again setting x = 0, we find $$c_2 = 0,$$ implying $$c_1 = 0$$ as well, thus the three vectors $$f_1,f_2,f_3$$ are linearly independent.