Linear maps: finding matrix

Homework Statement

Let T:R[x]2->R[x]3 be defined by T(P(x))=xP(x). Compute the matrix of T with respect to bases {1,x,x^2} and {1,x,x^2,x^3}. Find the kernel and image of T.

The Attempt at a Solution

I genuinely have no idea where to start on this, any pointers you can give me would be greatly appreciated.

lanedance
Homework Helper
i would start be computing the action of T on each basis vector of R2 & write in terms of basis of R3

then use that to make a matrix

notice the polynomials are considered as vectors, so for example ax2 +b x + c in the basis of R2 could just be written (c,b,a)

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so would the basis in R^2 in vector form be: {1,0,0},{0,1,0},{0,0,1}? If so, how would you get this in terms of the R^3 basis?

HallsofIvy
Homework Helper
so would the basis in R^2 in vector form be: {1,0,0},{0,1,0},{0,0,1}? If so, how would you get this in terms of the R^3 basis?
What you give, since they are in R3, form a basis for R3, not R2. A basis for R2 is {(1, 0), (0,1)}. (Don't use "{" and "}" for individual vectors. Those are set delimiters.)

lanedance
Homework Helper
so would the basis in R^2 in vector form be: {1,0,0},{0,1,0},{0,0,1}? If so, how would you get this in terms of the R^3 basis?

as halls points out its not R^2, I'm not too sure what the proper notation is, but as there are 3 independent basis vectors, it much more like R^3

so for the space, like R^3 with the basis {1,x,x2}, i think you're correct that you identify
(1,0,0) with 1
(0,1,0) with x & so on

simarlarly for the space where the image resides, with the basis {1,x,x2,x^3}, I would identify
(1,0,0,0) with 1
(0,1,0,0) with x & so on

is what you wrote exactly how the question is written?

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okay, so i have used the basis for R^3 and got a diagonal matrix with the elements x. Given the equation T(P(x))=x P(x) it looks like it could work. Is this correct? cheers

lanedance
Homework Helper
what matrix do you get, the way I'm thinking there shoudn't be any x's in the matrix?

if its from a 3 dimensional space to a 4 dimensional space, i think it should be a 4x3 matrix

Also i think it should have constant entries... consider what multiplying by x does, it shifts you from one basis vector to another... (similar to 90degree rotation in normal R^3)

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well i have just changed direction a bit but here is what i have so:
T(1,0,0)=(0,1,0,0); T(0,1,0)=(0,0,1,0); T(0,0,1)=(0,0,0,1)
So the matrix would be:
((0,0,0),(1,0,0),(0,1,0),(0,0,1))

(Each sub-bracket is a row)

lanedance
Homework Helper
sounds reasonable to me & all lines up with the initial definition