# Basis for Subspace

• Mar 21st 2007, 04:27 PM
Ranger SVO
Basis for Subspace
In the vector space of all real-valued functions, find a
basis for the subspace spanned by {sin(t); sin(2t); sin(t)cos(t)}

I can clearly see that sin(2t) can be written as 2*sin(t)cos(t), so that means that 2*sin(t)cos(t) and sin(t)cos(t) are linearly dependent sets.

Am I right so far?

So where do I go from here?
• Mar 21st 2007, 05:31 PM
topsquark
Quote:

Originally Posted by Ranger SVO
In the vector space of all real-valued functions, find a
basis for the subspace spanned by {sin(t); sin(2t); sin(t)cos(t)}

I can clearly see that sin(2t) can be written as 2*sin(t)cos(t), so that means that 2*sin(t)cos(t) and sin(t)cos(t) are linearly dependent sets.

Am I right so far?
Yup.

So where do I go from here?

Since sin(t) and sin(2t) are linearly independent it looks to me like your basis is {sin(t), sin(2t)} or {sin(t), sin(t)cos(t)}.

-Dan
• Mar 21st 2007, 05:48 PM
Ranger SVO
Quote:

Originally Posted by topsquark
Since sin(t) and sin(2t) are linearly independent it looks to me like your basis is {sin(t), sin(2t)} or {sin(t), sin(t)cos(t)}.

-Dan

I thank you for your response, but can it really be that simple?
Is it possible to have 2 basis for the same set?
• Mar 21st 2007, 05:51 PM
topsquark
Quote:

Originally Posted by Ranger SVO
I thank you for your response, but can it really be that simple?
Is it possible to have 2 basis for the same set?

Of course. Two possible bases for a 2D Euclidean space are the familiar:
{i, j} <-- Cartesian basis
{r, theta} <-- Plane polar basis
(where all of the above are unit vectors.)

-Dan
• Mar 21st 2007, 06:49 PM
ThePerfectHacker
Quote:

Originally Posted by topsquark
Since sin(t) and sin(2t) are linearly independent it looks to me like your basis is {sin(t), sin(2t)} or {sin(t), sin(t)cos(t)}.

I want to add to topsquarks post.
Since he does not explain why,
{sin(t),sin(t)cos(t)}
Those two a linearly independent (and hence for a basis for its subspace).

It is because these are not konstant multiples of each other.
Since,
sin(t)!=ksin(t)cos(t)

Using the special theorem for exactly two vectors.
• Oct 16th 2008, 12:00 PM
arpit
Are they really linearly independent??
are sin(t) and sin(t)*cos(t) really linearly independent???

because when t=0 both are zero and so is the linear combination and nt their scalar multiples...??
• Oct 16th 2008, 06:20 PM
ThePerfectHacker
Quote:

Originally Posted by arpit
are sin(t) and sin(t)*cos(t) really linearly independent???

because when t=0 both are zero and so is the linear combination and nt their scalar multiples...??

What you need to check is that if there exist $c_1,c_2$ so that $c_1 \sin t + c_2 \sin t \cos t = 0$ for all $t$ in some interval.