Question

It is linearly dependent bor Independent if f (x)=4
G(X)=x*2
h(x)=e*2x
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Answer 1

Answer:

If ex and e2x do happen to be linearly independent (which they are) it's impossible to assume that a1≠0, since the equation you wrote down would imply that a1=a2=0. This is where your reasoning fails.

To prove linear independence of (differentiable) functions, we usually want to integrate or differentiate. If we take the first derivative of a1ex+a2e2x=0, we get a1ex+2a2e2x=0. Subtracting the first equation from the second gives a2e2x=0, but since e2x≠0 for all x (in fact, we just need one point where it's nonzero) we see that a2=0. Substituting this then lets us show a1=0 as well.

Explanation:

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