This

**preview**shows page 1. to view the full**4 pages of the document.**Wednesday26/01/2011, Lecture notes byY. Burda

1Example of Changing Bases

Last time we’veseen the formulae

[v]A′=[I]A′

,A[v]A

[T]B′

,A′=[I]B′

,B[T]B,A[I]A,A′

In particular changing the basis in the domain of Tresults in performing

column-operations to its transformation matrix. Similarly,changing thebasis

in the range of Tresults in performing row-operations to its transformation

matrix.

Example:Find bases A′,B′of P2and P1so that the transformation

matrix [T]B′

,A′is in its reduced row-echelon form, where T:P2→P1,

T(a+bx +cx2)=(a−3b+c)+(2a−6b+3c)x

The idea is to start with arbitrary bases A,Bof P2and P1.With respect

to these bases thematrix of the transformation Tis likely not to bein its

row-reduced form. Wecan row-reduce it and ﬁnd amatrix Pso that P[T]B,A

is the row-reduced form of [T]B,A.Wecan then try to ﬁnd abasis B′such

that [I]B′

,B=P.Once suchabasis is found wewill get [T]B′

,A=[I]B′

,B[T]B,A

is in the reduced row-echelon form.

Let’s carry out this plan now:

Let A=(1,x, x2), B=(1,x)bethe standard bases of P2and P1.

Toﬁnd [T]B,Awecompute T(1) =1+2x,T(x)=−3−6x,T(x2)=1+3x

and put the coeﬃcients as columns of the transformation matrix:

[T]B,A=1−3 1

2−6 3

Now weshould reduce it using rowoperations and also makearecord of

the operations weare doing to ﬁnd the matrix Psuchthat P[T]B,Ais in

the row-reduced form. Wecan ﬁnd Pin asimpler way: multiplying any

matrix byPfrom the left is equivalentto performing on this matrix the row

operations that wedo to bring [T]B,Ato its row-reduced form. In particular

if weperform these operations on the identitymatrix, wewill get P.

1

www.notesolution.com

###### You're Reading a Preview

Unlock to view full version