Hi.
i need to prove that $\displaystyle \sin x$ and $\displaystyle \cos x$ are linearly independent.
what steps should i start with?
and how do i prove functions' linearly independent in general?
thanks in advanced!
Hi.
i need to prove that $\displaystyle \sin x$ and $\displaystyle \cos x$ are linearly independent.
what steps should i start with?
and how do i prove functions' linearly independent in general?
thanks in advanced!
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.
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.
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.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?