Question 29492: Let U and V be subspaces of Rn. Prove that the intersection, U n V, is also a subspace of Rn.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Let U and V be subspaces of Rn. Prove that the intersection, U n V, is also a subspace of Rn.
LET US USE THE FOLLOWING SYMBOLS...E TO SHOW ELEMENT OF.
LET U INTERSECTION V = W
1.0 E U AND 0 E V...SO 0 E W...SO W IS NOT EMPTY.
2.LET A,B BE ELEMENTS OF W AND X,Y BE SCALAR ELEMENTS IN THE FIELD OF RN.
3.THE ABOVE IMPLIES THAT A,B ARE ELEMENTS OF U .HENCE XA+YB IS AN ELEMENT OF U.
4.SIMILARLY.......XA+YB IS AN ELEMENT OF V.
5.HENCE XA+YB IS AN ELEMENT OF W.
6.THUS WE SHOWED THAT IF A,B ARE ELEMENTS OF W AND X,Y ARE ANY SCALARS IN THE FIELD OF RN THEN X+YB IS AN ELEMENT OF W.
7.HENCE W IS A SUBSPACE OF RN.
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