Linear Dependence Check: [-1, 1]

[AngeloG]

Check for Linear Dependence for: $$\sin \pi x$$ [-1, 1]

I'm thinking it's Linear Dependent. Since it says that any linear combination must be 0.

a*x + b*y = 0, a = b = 0.

So for any integer x, the value is 0. So [-1, 1] works.

 

[HallsofIvy]

? "Linear Dependence" or independence applies to a set of vectors. Certainly we can think of the collection of functions over [-1, 1] as a vector space but still $sin \pi x$ is a single function!

Also Linear Depence of a set of vectors does NOT mean "any linear combination must be 0". Only that there exist at least one more linear combination other than the one where all coefficients are 0. In order that two functions, f and g, be dependent, there must be a and b, not both 0 so that af(x)+ bg(x)= 0 for all x.

But still, what set of functions are you talking about? A single non-zero function (vector) is always independent.

 

[AngeloG]

The question is:

Check the linear dependency of the functions sin(pi x).

 

[Defennder]

You said "functions" but there's only one non-zero function. What kind of values can x take?

 

[AngeloG]

Err, it was part of:

1, cos(pi x), sin(pi x).

Those are the functions. 1 is linear independent, cos(pi x) and sin(pi x) I'm not sure about.

 

[HallsofIvy]

If you don't even know enough to quote the problem correctly, then I strongly recommend you review what "dependent" and "independent" mean! Once again, a single function (vector) is always "independent"! It makes no sense at all to say "1 is linearly independent" and, again, the problem is NOT asking about the "dependence" or "independence" of the each of those three functions. It is asking, as I suggested before, about the dependence or independence of the set of those three functions.

Now, how does your textbook define "dependent functions" or "dependent vectors"?