 #1
 153
 11
Summary:

We have set of functions ##\varphi (t)## continuous in the interval ##[a,b]##. The set is a linear (vector) space with the usual definitions of addition and multiplication by real numbers. We denote this space by ##C[a,b]##.
Statement of the problem : Prove that the following set of functions are linearly independent in the space ##C[a,b]## mentioned above : ##\varphi_1(t) = \sin^2 t, \varphi_2(t) = \cos^2 t, \varphi_3(t) = t, \varphi_4(t) = 3 \; \text{and} \; \varphi_1(t) = e^t##
Main Question or Discussion Point
We can make the first three functions add up to zero in the following way : ##\sin^2 t+\cos^2 t\frac{1}{3}\times 3 = \varphi_1(t) + \varphi_2(t)  \frac{1}{3} \varphi_3(t) = 0##.
However, look at ##\varphi_4(t) = t## and ##\varphi_5(t) = e^t##. How does one combine the two to add up to zero? I can't see a way out.
Any help would be appreciated.
However, look at ##\varphi_4(t) = t## and ##\varphi_5(t) = e^t##. How does one combine the two to add up to zero? I can't see a way out.
Any help would be appreciated.