Proving Linear Independence: Fixed t€R with {u,v}CR^2

[atakel]

Let t€R be fixed. Show that {u,v}CR^2 with u=(cost,sint), v=(-sint,cost) is a linearly inpedendent set.

 

[Pen_to_Paper]

My advice to you is to set up vectors u,v as a 2x2 matrix and plug in values for t;

then how do you conclude that a matrix has linearly independent vectors?

 

[atakel]

ı don't know ://

 

[micromass]

Do you know what "the rank" of a matrix is?

 

[atakel]

I uploadded the original question, there is no information about 'the rank'. In addition that I don't know how to solve the others.

 

[micromass]

So you've never heard of the rank of a matrix? Hmm, that makes it a bit more difficult.

Anyway, our u and v are independent iff

$$\alpha u+\beta v=0~\Rightarrow~\alpha=\beta=0$$

were alpha and beta are real number.
So, how do you proceed. You assume that there exists alpha and beta such that

$$\alpha (\cos t,\sin t)+\beta (-\sin t,\cos t)=(0,0)$$

this will give you a system of two equations and two unknowns (the alpha and beta).
Solve this system. If you find $$\alpha=\beta=0$$, then our u and v are independent.

 

[atakel]

thank u=)
if u have any idea about other questions, can u help me?
I didn't take a linear cource.. Solving these questions is really diffucult for me://

 

[micromass]

Yes, Id be happy to help you with these questions. But first you need to show me what you did yourself to try and solve these questions...

 

[atakel]

Actually, at first I started to study ''Partial Differential Equations in Action, Salsa'' but my backround is not enough to understand all subjects. Now I turned back and I started studying linear algebra..
For now, I don't have any idea about solving these questions. I have one week to learn all these subjects..

 

[micromass]

I'm sorry to say this, but you should first learn linear algebra and then start worrying abou th questions. We cannot help you until you learned some linear algebra

 

[atakel]

ok thank u:(

 

[micromass]

But hey, don't despair. If you're stuck on something, you can always ask us

 

[atakel]

=) I am going to finish all subjects asp.. anf then I will turn back with my new questions.. thank u very much=)