1. ## Linear Transformation

I have to solve the following problem:

Let V be the R-space of real polynomials of degree at most 3.
For p(X) in V, define

Tp(X)=1/6(19-6x+7x^2+32x^3-15x^4-6x^5)p'''(X)+(6-7x-9x^2+5x^3+x^4)p''(X)+(7-5x^2+2x^3)p'(X)+(10-6x^2)p(X).

Find [T]s where S is the standard ordered basis {1,x,x^2,x^3}.
Find [T]b where B is the non-standard ordered basis {2-12x-x^2+4x^3, 2-x^2, 3-x^2, -5x+2x^3}.

How do I work these out?

2. Originally Posted by PvtBillPilgrim
I have to solve the following problem:

Let V be the R-space of real polynomials of degree at most 3.
For p(X) in V, define

Tp(X)=1/6(19-6x+7x^2+32x^3-15x^4-6x^5)p'''(X)+(6-7x-9x^2+5x^3+x^4)p''(X)+(7-5x^2+2x^3)p'(X)+(10-6x^2)p(X).

Find [T]s where S is the standard ordered basis {1,x,x^2,x^3}.
Find [T]b where B is the non-standard ordered basis {2-12x-x^2+4x^3, 2-x^2, 3-x^2, -5x+2x^3}.

How do I work these out?
If $T:V\to W$
And $B=\{v_1,...,v_n\}$
And $B'=\{w_1,...,w_m\}$
Are the relative bases then,
$[T]_{B' B}=[[T(v_1)]_{B'}|[T(v_2)]_{B'}|...|[T(v_n)]_{B'}]$
Where the "||" are the adjoining the coloum vectors to the matrix.
That is the formula, the answer and solution might be long. These things get very computational. But that is the procedure that is used.

3. I think I need to assemble Tp(X) into a matrix. How do I do this?

4. Originally Posted by PvtBillPilgrim
I think I need to assemble Tp(X) into a matrix. How do I do this?
You find the transformations of the basis vectors.
Then you find their respective coordinate vectors relative to the second base.
Then the matrix obtained by the cololum coordinate vectors is the standard matrix for the linear transformation.
(That was the same thing I wrote only in nota
tion)

5. We took some simple example like y=(2/3)x and it became the matrices:

3 -2
2 , 3