Thread: How to show independent set of vectors

Consider the vector space of all continuous functions f : [-PI; P --> R with the
usual operations of adding two functions and multiplying a function by a scalar.
Show that the set of vectors (functions)
S = {sin(x); cos(x); sin(2x); cos(2x)}
is an independent set of vectors.

Originally Posted by flaming

Consider the vector space of all continuous functions f : [-PI; P --> R with the
usual operations of adding two functions and multiplying a function by a scalar.
Show that the set of vectors (functions)
S = {sin(x); cos(x); sin(2x); cos(2x)}
is an independent set of vectors.

to show that these functions are linearly independent, we must show that the only solution to is

Originally Posted by flaming

S = {sin(x); cos(x); sin(2x); cos(2x)}
is an independent set of vectors.

Define an inner product on - the space of all continous functions - by .

Now, is an orthogonal set of vectors.
You need to show,
1)
2)
3)
4)
5)
6)

But orthogonal sets are linearly independent. Therefore it follows this is linearly independent.

In general we can prove that is orthogonal (and so linearly independent) by extending the argument above.