# How to prove 2 functions are linearly independent

• Dec 26th 2012, 10:09 AM
Stormey
How to prove 2 functions are linearly independent
Hi.

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

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

• Dec 26th 2012, 10:49 AM
emakarov
Re: How to prove 2 functions are linearly independent
Suppose $\displaystyle a\sin x+b\cos x=0$ for all $\displaystyle x$. Substitute $\displaystyle x=0$ and $\displaystyle x=\pi/2$.
• Dec 26th 2012, 11:10 AM
Stormey
Re: How to prove 2 functions are linearly independent
if $\displaystyle x=0$ then $\displaystyle b=0$, and if $\displaystyle x=\frac{\pi}{2}$ then $\displaystyle a=0$, but what about every other x?
• Dec 26th 2012, 11:12 AM
emakarov
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?
• Dec 26th 2012, 12:48 PM
Stormey
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$\displaystyle \mathbb{R}^n$
and then if there exist scalars $\displaystyle a_0, a_1, .... a_n \in \mathbb{R}$ not all 0, so that $\displaystyle a_0\cdot v_0+a_1\cdot v_1 .... a_n\cdot v_n=0$ 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 $\displaystyle A\cdot \sin x+B\cdot \cos x=0$, but which x are we talking about here?
can you give me an example of two linear dependent functions? mayby that will help.
• Dec 26th 2012, 01:12 PM
emakarov
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 $\displaystyle \sin(x+\pi/3)$ and $\displaystyle \sin x+\sqrt{3}\cos x$. As with vectors, if f and g are linearly dependent, then f = a * g for some constant a.
• Dec 26th 2012, 03:14 PM
HallsofIvy
Re: How to prove 2 functions are linearly independent
Quote:

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.
• Dec 27th 2012, 02:00 AM
Stormey
Re: How to prove 2 functions are linearly independent
OK thanks for clearing that out for me.