Composition of Linear Transformations

[trap101]

Hi,

Two questions:

1) Compute the matrix product corresponding to the composition of the transformations. Let
U = P4(R) [polynomial degree 4], V = P3(R) , and W = P2, and let S = d/dx (derivative) and T = d/dx (derivative). Then the composition TS = d^2/dx^2 (second deriv)

Attempt: Now I'm assuming that we are using the standard basis vectors of the respective polynomials. So I apply the transformations to the respective polynomials. I forgot to mention S: U-->V , T: V-->W.

now applying the transformation to P4 I get: 4ax^3 + 3bx^2 + 2cx + d. From here I'm suppose to write this vector as a linear combination of the basis vectors, but my issue (if it is one) is that this is the only vector I have in P4 (trying to prove the general case), so am I safe to assume that the coeffcients that I get will be all that I have as my transformation vector from U-->V ? i.e: the coordinate vector? I thought I was suppose to get a full matrix? Once I get this part obviously the second part will follow. Help!



2) Let V = P3(R) and W = P4(R). Let D: W-->V be the derivative mapping D(p) = p', and let Int: V-->W be the integration mapping Int(p) = "integral sign" p(t) dt. Let "alpha" = {1,x,x^2,x^3} and "beta" = {1,x,x^2,x^3,x^4} be the standard bases in V and W. Compute transformation matrix "beta" to "alpha".

Attempt: Ok so I apply the transformation to the elements of "beta" individually. Now I get D(1) = 0, D(x) = 1, D(x^2) = 2x...etc. Now I wrote each individual transformation as a linear combination of the standard basis in "alpha" and I get a bunch of coordinate vectors, but one set of these coordinate vectors is all 0's. Is this the right approach or am I missing smoething?


Thanks for your guys help

 

[sunjin09]

You have 5 vectors in P4, i.e., x^4, x^3, x^2, x^1, x^0

 

[trap101]

I just realized my mistake, I meant to say "apply the transformation to P4, then I end up in P3"

 

[Fredrik]

You need to study the relationship between linear operators and matrices. It's explained e.g. in post #3 in this thread. (Ignore the quote and the stuff below it).

Edit: It's not clear to me what's meant by the transformation matrix from beta to alpha.