Prove that the list (x^3, sin(x), cos(x)) is linearly independent in V

[GluonZ]

Question: Prove that the list (x^3, sin(x), cos(x)) is linearly independent in V (V being the vector space of real-valued functions. In other words... common everyday math)

They're linearly independent, its pretty obvious. The issue is -- proving rigorously. This is not for an assignment, its for exam prep. (Friday)

Started by setting them as a linear combination:

c1*x^3 + c2*sin(x) + c3*cos(x) = 0 and trying to prove that ALL the coefficients need to be zero. But, nothing's rigorous enough.

Any ideas?

 

[matt grime]

If that is the zero function, then for all x, the LHS is zero. Why don'y you try putting some values for x in? The most obvious choice shows that c_3 is zero, thence it is trivial to find a value of x impliying that c_1 is zero, and then c_2 has to be zero as well.

 

[HallsofIvy]

Exactly. Although, instead of trying different values of x, I like staying with the easy x= 0:
Since these are functions
c1 x³+ c2 sin(x)+ c3 cos(x)= 0 means that it is 0 for all x.
Take x= 0 to get an easy equation. Of course it also follows that the derivative is 0 for all x. Differentiate and put x= 0 to get another easy equation. Differentiate a second time and set x= 0 to get a third easy equation.

 

[mathwonk]

what is the domain of your functions? if only one point, then they are not independent.

on the whole real line, just look at their zeroes to see sin and cos are independent. also if f depends on sin and cos, then its derivatives depend on their derivatives. this gives a contradiction eventually.