Show that g1 and g2 form a basis for V

[derryck1234]

Homework Statement

Let V be the space spanned by f₁ = sinx and f₂ = cosx.
(a) Show that g₁ = 2sinx + cosx g₂ = 3cosx form a basis for V.
(b) Find the transition matrix from B' = {g₁, g₂} to B = {f₁, f₂}.

Homework Equations

P = [[u'₁]_B [u'₂]_B...[u'_n]_B]

The Attempt at a Solution

I'm sorry but I don't know exactly how to start? Would the vectors f₁ and f₂ be represented in some form of coefficient matrix? Then how would that look? And then what would the vecors g₁ and g₂ look like in a coefficient matrix?

Please. I know this is not much of a solution...but any help would go a long way.

Thanks

Derryck

 

[DiracRules]

What is the definition of a space spanned by two vectors?

If you find this difficult, try to answer to this question:
Let's take $R^2$. It is spanned by two vectors, for example $\left[\begin{array}{c}1\\0\end{array}\right],\left[\begin{array}{c}0\\1\end{array}\right]$.
How can you describe each element of $R^2$ in terms of these vectors?

If you answer to this question, it is not difficult to understand HOW is built V.

 

[derryck1234]

Ok, so to represent any element in R2 I would say (a, b) = t(1, 0) + v(0, 1)

Are you then saying that (a, b) = cosx(1, 0) + sinx(0, 1)?

 

[DiracRules]

You're almost there.

Saying that $R^2=span\{e_1=\left[\begin{array}{c}1\\0\end{array}\right],e_2=left[\begin{array}{c}0\\1\end{array}\right]\}$ , means that, as you wrote,
[item]\forall \mathbf{v}\in R^2,\,\, \mathbf{v}=\alpha\cdot e_1+\beta\cdot e_2,\,\,\alpha,\beta\in R[/item]

In this case, the problem says that f_1=sin(x) and f_2=cos(x) span the space V. Hence, you can write that [item]\forall \mathbf{v}\in V,\,\, \mathbf{v}=\alpha\cdot f_1+\beta\cdot f_2=\alpha\cdot\sin(x)+\beta\cdot\cos(x)[/item]

Now you should be on the right way...

 

[derryck1234]

um? your R^2=SPAN/{E_1/LEFT[...is not turning into the nice representation you had previously accomplished? I'm not quite understanding:( I'm really sorry DiracRules...

 

[DiracRules]

Sorry, maybe some problem with the forum.

However, the vectors e_1=[1;0] and e_2=[0;1] spans R^2 means that every element of R^2 is a linear combination of e_1 and e_2, that is, you can write it as v=a e_1+b e_2.

In you exercise, it says that the vectors f_1=sin(x) and f_2=cos(x) span the space V.
What does this mean? How can you describe every element of your space?

 

[derryck1234]

By V = asinx + bcosx...I knew/thought of this approach at the beginning. But I don't understand how I would prove that g1 and g2 form a basis for the same space?

 

[DiracRules]

You have to show that every element of V can be written in terms of g_1 and g_2.
If v is an element of V, you can write, as we said, v=a f_1+b f_2=a sin(x)+b cos(x).
Try to do the same thing with g_1 and g_2 and see if you can get to the same form.

To answer to the last question, try to think at f_1,f_2,g_1 and g_2 as two dimensional vectors: f_1=(sin(x), 0) f_2=(0, cos(x) ) g_1=... g_2=...
This way you can find (through simple observations or trial-and-error :P ) the transition matrix.

 

[derryck1234]

Ok. Now we getting somewhere:) I'm sorry for actually being so stupid! But I have no lecturer since this is a correspondence course I am doing...

Yes...and I worked through it now and got the right answer in the textbook:)

Thanks a lot Diracrules!

Ciao

 

[DiracRules]

Do not worry... just think easily and all will go well :D