How to prove 2 functions are linearly independent

Hi.

i need to prove that and are linearly independent.
what steps should i start with?

and how do i prove functions' linearly independent in general?

thanks in advanced!

Suppose for all . Substitute and .

if then , and if then , but what about every other x?

Can you formulate what it means for sin(x) and cos(x) to be linearly independent?

that's the thing, i'm not sure.
i only know how to check for liniar dependence when it is vectors from
and then if there exist scalars not all 0, so that then the vectors linear dependent.
but here with this x - i can't get the hang of it.
i get that i need to show that there aren't A and B not all 0 so that , but which x are we talking about here?
can you give me an example of two linear dependent functions? mayby that will help.

The definition of linearly independent functions is the same as for vectors, but the issue is in the definition of +, *, = and 0. Here + denotes pointwise addition of functions, multiplication is pointwise multiplication of a function by a constant, and equality is equality of functions, not numbers. When we say that f = g, it means that f(x) = g(x) for all x (this definition of function equality is sometimes called extensional because it depends only on the functions' output and not their algorithms). Finally, 0 is the function that returns the number 0 for all arguments. So, in proving that sin(x) and cos(x) and linearly independent, we assume that there exist a and b such that a * sin(x) + b * cos(x) = 0 for all x. Then we need to show that a = b = 0, which you have done.

An example of two linearly dependent functions is and . As with vectors, if f and g are linearly dependent, then f = a * g for some constant a.

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i get that i need to show that there aren't A and B not all 0 so that , but which x are we talking about here?

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emakarov answered this in his very first response: when we talk about two functions, f and g, being linearly independent we mean that the only a and b such that af(x)+ bg(x)= 0 for all x are a= b= 0. In particular, if a sin(x)+ bcos(x)= 0 for all x, then, for x= [itex]\pi/2[/itex], we have a(1)+ b(0)= a= 0. Now we know that bcos(x)= 0 for all x. Either b= 0 or cos(x)= 0 for all x.

OK thanks for clearing that out for me.
that was very helpful!