Interesting Linear Transformation question

• Nov 2nd 2010, 11:13 AM
jayshizwiz
Interesting Linear Transformation question
*** $U \text{ and } W \in R_4[x]$
and below I simply transform from $R_4[x] \rightarrow R^4$

$W = (i,j,0,-j) \phantom{hacklol} U = (4s, 4t, -s, -t)$

Does there exist a linear transformation $T:R_4[x] \rightarrow R_4[x]$ so that $T(U)=W$ and $T(W) = U$

So I got to this...

$W = Sp{(1,0,0,0),(0,1,0,-1)}$
$U = Sp{(4,0,-1,0),(0,4,0,-1)}$

And if I prove it for the bases, I prove it for the entire thing. But how do I continue from here?

I know in general that if you have a basis you can create any transformation you want but I don't know how to.

Thanks!
• Nov 2nd 2010, 03:18 PM
jayshizwiz
Ok, I think I got it. It took me many pages on my notebook...

Basically, $T(x,y,z,r) = (4x+15z, -5r-y, -x-4z, r)$

Thanks anyway!

***and after that I must change it back to R_4[x]...
• Nov 2nd 2010, 06:41 PM
Drexel28
Quote:

Originally Posted by jayshizwiz
$W = (i,j,0,-j) \phantom{hacklol} U = (4s, 4t, -s, -t)$

Does there exist a linear transformation $T:R_4[x] \rightarrow R_4[x]$ so that $T(U)=W$ and $T(W) = U$

So I got to this...

$W = Sp{(1,0,0,0),(0,1,0,-1)}$
$U = Sp{(4,0,-1,0),(0,4,0,-1)}$

And if I prove it for the bases, I prove it for the entire thing. But how do I continue from here?

I know in general that if you have a basis you can create any transformation you want but I don't know how to.

Thanks!

What exactly are these spaces?
• Nov 3rd 2010, 01:51 AM
jayshizwiz
Oy! Sorry, I always do that. I try to save time so I just give half the question... Maybe later, if I have more time, I'll just post the entire question with my answer and hopefully itwill be right.

I edited it at the top of my original post. Both spaces are in R_4[x]. Sorry for the confusion
• Nov 3rd 2010, 08:49 AM
Drexel28
Quote:

Originally Posted by jayshizwiz
Oy! Sorry, I always do that. I try to save time so I just give half the question... Maybe later, if I have more time, I'll just post the entire question with my answer and hopefully itwill be right.

I edited it at the top of my original post. Both spaces are in R_4[x]. Sorry for the confusion

Haha, the funny this is my problem was not knowing what $R_4[x]$ is? Maybe $\mathbb{R}_4[x]=\left\{a_0+\cdots+a_4x^4:a_0,\cdots,a_4\in\mathbb {R}\right\}$ but from the elements I would assume not.
• Nov 3rd 2010, 01:45 PM
jayshizwiz
$R_4[x] = a_0 + a_1x+a_2x^2+a_3x^3 : a_0,...,a_3 \in R$