Let C(R) be the vector space of continuous real functions over the field R (wher

Question

Let C(R) be the vector space of continuous real functions over the field R (where R denotes the real numbers). Then we want to prove that {sin, cos, exp,p0} is a linearly independent set where exp=e^x and p0=1 for all x in R, but I need to prove this using the 'classic' definition of linear independence and not the Wronskian. That is, I want to show that acos(x)+bsin(x)+ce^x+d(1)=0, where 0 denotes the zero function in R if and only if a=b=c=d=0. I can't think of a way to do this, please help I will rate! Let C(R) be the vector space of continuous real functions over the field R (where R denotes the real numbers). Then we want to prove that {sin, cos, exp,p0} is a linearly independent set where exp=e^x and p0=1 for all x in R, but I need to prove this using the 'classic' definition of linear independence and not the Wronskian. That is, I want to show that acos(x)+bsin(x)+ce^x+d(1)=0, where 0 denotes the zero function in R if and only if a=b=c=d=0. I can't think of a way to do this, please help I will rate!

Explanation / Answer

Pf. We have the vectors
[sin(x), cos(x), e^x,    1]. To test linear independence, we write
Asin(x) + Bcos(x) + Ce^x + D = 0
and hold it true for ALL x. Setting x = 0, we immediately get B = -C - D. However, we note that if x went to infinity, the limiting behavior of e^x would imply that A, B, C, and D all equal zero.

Choose x = infinity, then

Asin(inf) + Bcos(inf) + Ce^(inf) + D = 0 iff A = B = C = D = 0.

*Recall sine and cosine are written as (1/2i)[e^(ix) - e^(-ix)] and (1/2)[e^(ix) - e^(-ix)]
Therefore the vectors form a linearly independent set. I hope this helps; I completely missed the "no-wronskian" clause.
Therefore the vectors form a linearly independent set. I hope this helps; I completely missed the "no-wronskian" clause.

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