Matrices of linear transformations

derryck1234
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Homework Statement



Let T: P2 - P2 be the linear operator defined by
T(a0 + a1x + a2x2) = a0 + a1(x - 1) + a2(x - 1)2

(a) Find the matrix for T with respect to the standard basis B = {1, x, x2}.

Homework Equations



[T]B[x]B = [T(x)]B

The Attempt at a Solution



T(1) = a0 + a1(1 - 1) + a2(1 - 1)2
= a0

Okay...I don't know if I'm going in the right direction here?...and if I am, I'm not sure how to obtain vectors from the proceeding expressions for T(x) and T(x2)??

Any help would be appreciated.

Thanks.

Derryck.
 
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Hi Derryck! :smile:
derryck1234 said:
Let T: P2 - P2 be the linear operator defined by
T(a0 + a1x + a2x2) = a0 + a1(x - 1) + a2(x - 1)2

(a) Find the matrix for T with respect to the standard basis B = {1, x, x2}.
…
T(1) = a0 + a1(1 - 1) + a2(1 - 1)2
= a0

nooo :redface: …

suppose it was T(a0 + a1x + a2x2) = a0 + a1(x + 1) + a2(x + 1)2

then your method would give you T(1) = a0 + a1(1 + 1) + a2(1 + 1)2
= a0 + 2a1 + 4a2

but the correct way is to start by saying 1 = 1 + 0x + 0x2, so T(1) = … :wink:

carry on from there :smile:​
 
Wo...?

But then is this the same as saying (in vector form):

1 1
0 = 0
0 0 ?

I'm not sure what you mean by 1 = 1 + 0x + 0x2, because what I see from this, I would think that it would be more correct to say:

1 = a0 + a1x + a2x2, then to solve...? Flip, I'm so bad at this I know...but please, just humor me a little...it will help a great deal...

Cheers

Derryck
 
derryck1234 said:
I'm not sure what you mean by 1 = 1 + 0x + 0x2, because what I see from this, I would think that it would be more correct to say:

1 = a0 + a1x + a2x2, then to solve...?

Yes, and the solution is a0 = 1, a1 = a2 = 0, isn't it? :wink:

In other words: in the basis {1,x,x2}, 1 has components (1,0,0).

ok, now what are the components of x ?

and what are the components of x2 ? :smile:​
 
Ok, the components of x would be found by solving:

x = a0 + a1(x - 1) + a2(x - 1)2

So I would then say

a1(x - 1) = x
Therefore a1 = x / (x - 1)... Am I going along the right lines? I'm not too sure what to do from here.

Thanks for the help so far though really!

Cheers

Derryck
 
derryck1234 said:
Ok, the components of x would be found by solving:

x = a0 + a1(x - 1) + a2(x - 1)2

No!

In the basis {1,x,x2}, what is x?

x = ?1 + ?x + ?x2
 

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