Question 62609
Consider the linear transformation T : R3 -> R2 whose matrix
with respect to the standard bases is given by
LET T=
=
2 1 0
0 2 -1

Now consider the bases:
f1= (2, 4, 0)
f2= (1, 0, 1)
f3= (0, 3, 0) of R3 and
g1= (1, 1)
g2= (1,−1) of R2
Compute the coordinate transformation matrices between the standard
bases and these bases 
F IS THE MATRIX OF BASE VECTORS f1,f2,f3
= 
2,1,0
4,0,3
0,1,0

MATRIX OF STANDARD BASE IN R3 IS E3 SAY 
=
1,0,0
0,1,0
0,0,1

G IS THE MATRIX OF BASE VECTORS g1,g2.
=
1,1
1,-1

MATRIX OF STANDARD BASE IN R2 IS E2 SAY
=
1,0
0,1

F^(-1) IS [HOPE YOU KNOW HOW TO INVERT MATRIX.OTHERWISE PLEASE COME BACK]
=
0.5,0,-0.5
0,  0,  1
-2/3,1/3,2/3
HENCE TRANSFORMATION MATRIX FROM E BASIS TO F BASIS IS
 (XF)=[F^(-1)]*(XE)  WHERE XF REPRESENTS X IN F BASIS.ETC..AND 
IF YE = A*(XE)......THEN
(YF) = [F^-1]*(YE)

G^-1 IS  
=
0.5,0.5
0.5,-0.5
SIMILARLY AS ABOVE,WE HAVE
TRANSFORMATION MATRIX FROM E BASIS TO G BASIS IS
 (XG)=[G^(-1)]*(XE)  WHERE XG REPRESENTS X IN G BASIS.ETC..AND 
IF (YE) = A(XE)......THEN
(YG) = [G^-1]*(YE)
--------------------------------------------------------------------------
and compute the matrix of T with respect to the new
bases.
HENCE 
IF TE IS IN E BASIS
(TF)= [F^(-1)]*T*F
TE is given 
=
2,0
1,2
0,-1

SIMILARLY IN G BASIS,TAKING T AS GIVEN, WE GET
TG=[G^(-1)]*T*G

IT IS NOT CLEAR FROM YOUR PROBLEM WHETHER YOU ARE WRITING VETICAL VECTORS AS HORIZONTAL VECTORS FOR CONVENIENCE.PLEASE CLARIFY.IN SUCH A CASE BETTER WRITE AS (x1,x2,x3)'
on your feed back we can complete the solution or you can continue using the above formula

Any help would be appreciated. Thank you!