# Thread: Linear Transformation and Matrices

1. ## Linear Transformation and Matrices

Let L: P1 --> P2 be define be L(p(t)) = tp(t) + p(0). Consider the ordered bases S={t,1} and S'={t+1, t-1} for P1 and T={t2, t, 1} and T'={t2+1, t-1, t+1} for P2. Find the representation of L with respect to:
a) S and T
b) S' and T'
c) Find L(-3t-3) by using the definition of : and the matrices obtained in parts (a) and (b).

2. I honestly have no idea where to start with this. I understand the simple examples that they provide in the books, but I am not sure how to do this...

3. first, define the images under L of the basis vectors of S, and express these as linear combinations of the basis T. from this, the matrix for L w.r.t. these bases should be clear. then do the same for S' and T'.

i'll get you started:

L(1) = t = (0)t^2 + (1)t + (1)1
L(t) = t^2 = (1)t^2 + (0)t + (0)1.

in other words L(p(t)) = L((a1,a2)S) = L(a1(t) + a2(1)) = a1(L(t)) + a2(L(1)) = a1(t^2 + 0t + 0) + a2(0t^2 + t + 1) = (a1t^2 + a2t + a2) = (a1,a2,a2)T

part (a) will be the easy part, because S and T are the normal bases you expect for P1 and P2.

part (b) will be trickier because the S'-coordinates for at + b are not (a,b), and the T'-coordinates for ct^2 + dt + e are not (c,d,e).