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Examples of change of basis, invertable functions, invertable transformations

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Martin, Burda
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Wednesday26/01/2011, Lecture notes byY. Burda
1Example of Changing Bases
Last time we’veseen the formulae
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
Example:Find bases A,Bof P2and P1so that the transformation
matrix [T]B
,Ais in its reduced row-echelon form, where T:P2P1,
T(a+bx +cx2)=(a3b+c)+(2a6b+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 Bsuch
that [I]B
,B=P.Once suchabasis is found wewill get [T]B
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)=36x,T(x2)=1+3x
and put the coefficients as columns of the transformation matrix:
[T]B,A=13 1
26 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.
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