Linear Independence: Showing 1, sin^2(x), sin(2x) is Independent

[skoomafiend]

Homework Statement

there is the vector space F(R) = {f | f:R -> R }
show that {1, sin^2(x), sin(2x)} is linearly independent

Homework Equations

a(1) + b(sin^2(x)) + c(sin(2x)) = 0, where the ONLY solution is a=b=c=0, for the set to be implied linearly independent.

The Attempt at a Solution

for that set to be considered linearly independent, it has to be linearly independent (a=b=c=0) for ALL values of x?

i mean, for x = 0

a(1) + b(sin^2(x)) + c(sin(2x)) = 0

0(1) + 1(0) + 1(0) = 0, and that would be a linearly dependent set since not all coefficients are 0.

but that is only one case. do i have to show that this is not valid for EVERY case? what would be a good way to approach these types of problems?

Thanks!

 

[Mark44]

Homework Statement

there is the vector space F(R) = {f | f:R -> R }
show that {1, sin^2(x), sin(2x)} is linearly independent

Homework Equations

a(1) + b(sin^2(x)) + c(sin(2x)) = 0, where the ONLY solution is a=b=c=0, for the set to be implied linearly independent.

The Attempt at a Solution

for that set to be considered linearly independent, it has to be linearly independent (a=b=c=0) for ALL values of x?

Yes. For the three functions to be linearly independent, the equation a(1) + b(sin^2(x)) + c(sin(2x)) = 0 hold for all values of x, and the only solutions for the constants must be a = b = c = 0.

i mean, for x = 0

a(1) + b(sin^2(x)) + c(sin(2x)) = 0

0(1) + 1(0) + 1(0) = 0, and that would be a linearly dependent set since not all coefficients are 0.

but that is only one case. do i have to show that this is not valid for EVERY case? what would be a good way to approach these types of problems?

Thanks!

 

[skoomafiend]

what would be a valid way to show that the set is linearly independent?