# Thread: How to prove 2 functions are linearly independent

1. ## How to prove 2 functions are linearly independent

Hi.

i need to prove that $\sin x$ and $\cos x$ are linearly independent.

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

2. ## Re: How to prove 2 functions are linearly independent

Suppose $a\sin x+b\cos x=0$ for all $x$. Substitute $x=0$ and $x=\pi/2$.

3. ## Re: How to prove 2 functions are linearly independent

if $x=0$ then $b=0$, and if $x=\frac{\pi}{2}$ then $a=0$, but what about every other x?

4. ## Re: How to prove 2 functions are linearly independent

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

5. ## Re: How to prove 2 functions are linearly independent

that's the thing, i'm not sure.
i only know how to check for liniar dependence when it is vectors from $\mathbb{R}^n$
and then if there exist scalars $a_0, a_1, .... a_n \in \mathbb{R}$ not all 0, so that $a_0\cdot v_0+a_1\cdot v_1 .... a_n\cdot v_n=0$ 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 $A\cdot \sin x+B\cdot \cos x=0$, but which x are we talking about here?
can you give me an example of two linear dependent functions? mayby that will help.

6. ## Re: How to prove 2 functions are linearly independent

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 $\sin(x+\pi/3)$ and $\sin x+\sqrt{3}\cos x$. As with vectors, if f and g are linearly dependent, then f = a * g for some constant a.

7. ## Re: How to prove 2 functions are linearly independent

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?
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= $\pi/2$, 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.

8. ## Re: How to prove 2 functions are linearly independent

OK thanks for clearing that out for me.