Thread: 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?

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

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?

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

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.

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...??

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 so that for all in some interval.