Editing Linear independence (section)

== Linear independence of functions ==

Let <math>V</math> be the [[vector space]] of all differentiable [[function (mathematics)|function]]s of a real variable <math>t</math>. Then the functions <math>e^t</math> and <math>e^{2t}</math> in <math>V</math> are linearly independent.

=== Proof ===
Suppose <math>a</math> and <math>b</math> are two real numbers such that

:<math>ae ^ t + be ^ {2t} = 0</math>

Take the first derivative of the above equation:

:<math>ae ^ t + 2be ^ {2t} = 0</math>

for {{em|all}} values of <math>t.</math> We need to show that <math>a = 0</math> and <math>b = 0.</math> In order to do this, we subtract the first equation from the second, giving <math>be^{2t} = 0</math>. Since <math>e^{2t}</math> is not zero for some <math>t</math>, <math>b=0.</math> It follows that <math>a = 0</math> too. Therefore, according to the definition of linear independence, <math>e^{t}</math> and <math>e^{2t}</math> are linearly independent.