Matrix representation

1. Jan 26, 2009

jeffreylze

1. The problem statement, all variables and given/known data

Let T: P2 > P3 denote the function defined by multiplication by x :T(p(x)) = xp(x). In other words, T(a+bx+cx2) = ax+bx2+cx3

(a) Show that T is a linear transformation.
(b) Find the matrix of T with respect to the standard bases {1,x,x2} for P2 and {1,x,x2,x3} for P3

2. Relevant equations

3. The attempt at a solution

I managed to prove that T is a linear transformation. With B, I have completely no idea how to go about, i checked my books but the examples given are not relevant. Please help.

2. Jan 26, 2009

Dick

Figure out where the standard basis vectors map. 1 in P2 is (1,0,0) in the given basis. That maps to x in P3, which is (0,1,0,0). Continue this for all of basis vectors in P2, scratch your head and figure out how to write a matrix which does the same thing. THINK about it.

3. Jan 26, 2009

jeffreylze

I dont really understand this part, how does 1 in p2 maps to x in p3 ?

4. Jan 26, 2009

Dick

T(p(x))=x*p(x). So if p(x)=1, then T(1)=x*1=x. 1 in P2 maps to x in P3.

5. Jan 26, 2009

jeffreylze

Oh, so

p(x) = x , x in p2 will map it to x2 in p3? - (0,0,1,0)
p(x) = x2, X2 in p2 will map it to x3 in p3 - (0,0,0,1)

So that gives me

A = [0 0 0;
1 0 0
0 1 0;
0 0 1] ?

But is there a faster way to do this? I came across this equation while looking for extra info online, [T(u)]C=AB . Will that simplify the method?

6. Jan 26, 2009

Dick

That's exactly right. What was "not faster" about the way you already did it? Once you understood the problem you solved it in two minutes. Why complicate it?

7. Jan 26, 2009

jeffreylze

Using that method, i tried solving this question but to no avail :

Find the matrix representation of T:P1 > P2 with respect to bases B = {1,x} and C {1,x,x2} where

T(p) = (x+2)p for p$$\in$$P1

p = a0 + a1x

T(1) = (x+2)
T(x) = (x2+2x)

and I dont know how to map that to P2

8. Jan 26, 2009

Dick

1 in P2 is (1,0). (x+2) in P2 is (2,1,0). PLEASE say you knew that. What is x in P1 and what is (x^2+2x) in P2? Now write down a 2x3 matrix and start filling in the columns. Your best tool is actually thinking about the problem. There is no magic formula.

9. Jan 26, 2009

jeffreylze

I am such a douche + a slow learner =/ Yeah, i get it (x+2) in P2 is (2,1,0) (it is just the same thing like the previous example! careless me) x in p1 is just (0,1) so (x^2+2x) in P2 will be (0,2,1). So the matrix will be A = (2,1,0 ; 0,2,1)

So a different basis will still be the same. Say B = {1,x-2} and C = {2,x,x^2} for P1 and P2 respectively.

T(1) = (x+2)
T(x-2)= (x^2-4)

1 in P1 will be (1,0) , and will map to (1,1,0) in P2

(x-2) in P2 will be (0,1) and will map to (-2,0,1) in P2

Yeap, eureka, i think i got it. Also, do you think these steps will be sufficient to answer exams questions? Or do I need a more rigid method/calculations ?

10. Jan 27, 2009

Dick

You are doing exactly what you are supposed to do. Compute the product, figure out the components in the given bases and deduce the matrix.