Question 36941: Question about linear transformation proofs. For T: V-->W in R2, v in V, T(v)=T(x2,y2)=(x2^5, y2^5), and I need to use "non-standard" vector operations for addition and scalar multiplication(such as, u=(x1,y1),v=(x2,y2), and u+v=(x1^5 + y1^5, x2^5 + y2^5))in V, but I need to use "standard" vector addition and scalar multiplication in W. I can't figure out how to use the standard operation in W for the proofs. Can't I just use the operations in V to show that the T is a linear transformation? Or, should I show proofs in both V and W separately, using the different vector operations? Please help me!!
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! YOUR QUESTION IS NOT COMPLETE,BUT FROM WHATEVER IS GIVEN ,I TAKE IT AS FOLLOWS
1.YOU ARE GIVEN A VECTOR SUBSPACE IN T WITH ADDITION AND MULTIPLICATION DEFINED BY SPECIAL OPERATIONS....OK....THEN WE HAVE TO USE ONLY THOSE SPECIAL OPERATIONS TO FIND SUM OR PRODUCT IN THIS SUB SPACE..FOR EX.IF DEFINITIONS ARE AS FOLLOWS THEN
t1=(X1,Y1) t2=(X2,Y2)
t1+t2=[{X1^(1/3)+X2^(1/3)}^3,{Y1^(1/3)+Y2^(1/3)}^3]
C*t1=(C^3*X1,C^3*Y1) ETC...
WE FOLLOW THOSE DEFINITIONS TO DO OPERATIONS IN T..
IT IS ASSUMED THAT THOSE OPERATIONS ARE CLOSED ,ASSOCIATIVE ETC...IN ACCORDANCE WITH REQUIREMENTS OF VECTOR SUBSPACES OR WE NEED TO PROVE IT.
2.IF T MAPS ON TO W OR IS TRANSFORMED IN TO W ,TO GET THE TRANSFORMATION INTO W FROM T WE USE THE ABOVE OPERATIONS.
THAT IS IF AS GIVEN ABOVE...t1+t2=w1 =(P1,Q1) SAY THEN WE TAKE
t1+t2=[{X1^(1/3)+X2^(1/3)}^3,{Y1^(1/3)+Y2^(1/3)}^3] =w1=(P1,Q1)
OR...P1={X1^(1/3)+X2^(1/3)}^3 AND Q1 ={Y1^(1/3)+Y2^(1/3)}^3...ETC...
3. BUT ONCE WE ARE IN W SPACE ,AS STIPULATED IN THE PROBLEM,
IF W1=(P1,Q1),W2=(P2,Q2)....WE TAKE IT THAT
W1+W2=(P1+P2,Q1+Q2)...ETC....
HOPE IT IS CLEAR AND THIS IS WHAT YOU WANTED.
|
|
|
