Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 21 September 2014

Lecture 11: [t]m[t] and multiplication by t

In the same way that m is with m=0 0 111 102 93 84 75 6 [t]m[t] is [t] with m=0.

Let m=(t-2)2=t2-4t+4. Then, in [t](t-2)2[t], t-2-4t+4=0, t2=4t-4, and t3+7t2+5t+3 = t·t2+7t2+5t+3 = t(4t-4)+7 (4t-4)+5t+3 = 4t2-4t+28t-28+5t+3 = 4(4t-4)-4t+28t +5t-25 = 16t-16+29t-25 = 45t-41. Any polynomial in [t](t-2)2[t] is a linear combination of 1 and t. B={1,t}is a basis of [t](t-2)2[t]. C={1,t-2}is another basis of [t](t-2)2[t], and the change of basis matrix from B to C is P= ( 1-2 01 ) . Let V=[t](t-2)2[t] and let f:VV be the linear transformation given by f(p)=tp. For example: f(45t-41) = t(45t-41) = 45t2-41 = 45(4t-4)-41 = 180t-180-41 = 180t-221. The matrix of f with respect to B is Bf= ( 0-4 14 ) since f(t)=t2=4t-4. The matrix of f with respect to C is Cf= ( 20 12 ) .

V=[t](t-2)(t-3)[t] has (t-2)(t-3)=0, so that t2-5t+6=0 and t2=5t-6. V has bases B={1,t} and C={t-2,-t+3}. Let f:VV be the linear transformation given by f(p)=tp. The matrix of f with respect to B is Bf= ( 0-6 15 ) . The matrix of f with respect to C is Cf= (3002) since f(t-2) = t2-2t= 5t-6-2t= 3t-6=3(t-2), f(-t+3) = -t2 +3t=-(5t-6)+ 3t=-2t+6=2(-t+3).

V = [t](t-3)[t] [t](t-2)[t] = { (u,w)| u[t](t-3)[t], w[t](t-2)[t] } and t·(u,w)=(tu,tw). V has basis C={(1,0),(0,1)} with t(1,0) = (t,0)= (3,0)= 3(1,0), t(0,1) = (0,t)=(0,2) =2(0,1) since t=3 in [t](t-3)[t] and t=2 in [t](t-2)[t]. So the matrix of f:VV given by f(u,w)=t(u,w), is Cf= (3002). The matrix of f with respect to the basis B={(1,1),(t,t)}= {(1,1),(3,2)} is B=(0-615) since t(3,2)= (3t,2t)= (9,4)= -6(1,1)+5(3,2). The function Φ: [t](t-2)(t-3) [t](t-3)[t] [t](t-2)[t] given by Φ(1)=(1,1) andΦ(t)= (t,t)=(3,2) is a linear transformation such that if v[t](t-2)(t-3) then Φ(tv)=tΦ(v).

HW: Show that Φ is bijective.

Notes and References

These are a typed copy of Lecture 11 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on August 17, 2011.

page history