Question 62609: Consider the linear transformation T : R3 -> R2 whose matrix
with respect to the standard bases is given by
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 and compute the matrix of T with respect to the new
bases.
Any help would be appreciated. Thank you!
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! 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!
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