I have the following set of functions: $$f_1(x) = 5x$$ $$f_2(x) = 2x+3$$ $$f_3(x) = 2x^2 + 3$$ $$f_4(x) = x^2+ x$$
I need to find $C_1, C_2, C_3, C_4$ which make the linear combination equal to zero for all x.
I have tried many ways that I can think of. For example, I tried solving for $\mathbf x = \mathbf 0 * \mathbf Ainverse$ but couldn't do the gauss eliminatin to find $\mathbf Ainverse$. I proceeded to write out the linear combination and group the coefficients but I get 3 equations and 4 unknowns so that doesn't work either. Could someone give me a push in the right direction? Any advice is greatly appreciated. thank you.
For the second method I mentioned I have the following: $$(3*C_2 + 3*C_3) + (5*C_1 + 2*C_2 + C_4)x + (2*C_3 + C_4)x^2 = 0$$ This becomes 3 equations with 4 unkowns, which I cannot solve for.