1
, Problem 1
We have
= ( − , ), = ( , − ), which form a basis of .
( ) = ( , − , ), ( ) = ( , − , ).
Step : Express ( , ) as a linear combination of , :
Let
( , )= (− , )+ ( ,− )
That gives:
− + =?
Actually solve:
− + =
− =
Multiply first by :
− + =
Add to second:
(− + )+( − )= +
So = + .
Then from first equation:
− + ( + )=
− + + =
− =− −
= + .
Thus:
( , )=( + ) +( + )
2
, Problem 1
We have
= ( − , ), = ( , − ), which form a basis of .
( ) = ( , − , ), ( ) = ( , − , ).
Step : Express ( , ) as a linear combination of , :
Let
( , )= (− , )+ ( ,− )
That gives:
− + =?
Actually solve:
− + =
− =
Multiply first by :
− + =
Add to second:
(− + )+( − )= +
So = + .
Then from first equation:
− + ( + )=
− + + =
− =− −
= + .
Thus:
( , )=( + ) +( + )
2