Question 26471: Find the dimension os span ( (0,-1,k), (1,-1,0), (1,1,1) ) for each real k.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Find the dimension os span ( (0,-1,k), (1,-1,0), (1,1,1) ) for each real k.
LET US FIND THOSE VALUES OF K WHICH WILL MAKE (0,-1,K)=A(1,-1,0)+B(1,1,1)
A+B=0........I
-A+B=-1.......II
B=K ............III
EQN.I+EQN.II
2B=-1
B=-1/2
HENCE A=1/2
K=B=-1/2
HENCE FOR K=-1/2,THE SPACE WILL BE SPANNED BY BASIS OF 2 INDEPENDENT ELEMENTS
(1,-1,0)AND (1,1,1)...THE NUMBER OF INDEPENDENT ELEMENTS IN THE BASIS IS CALLED THE DIMENSION.PLEASE NOTE THAT BASIS SHOULD CONSIST OF ONLY INDEPENDENT ELEMENTS.THERE ARE 2 ELEMENTS NAMELY... (1,-1,0)AND (1,1,1)...HERE.SO THIS
HAS DIMENSION OF 2.
FOR OTHER VALUES OF K , WE SHALL HAVE A SPACE SPANNED BY BASIS OF 3 GIVEN INDEPENDENT ELEMENTS.
(0,-1,k), (1,-1,0), (1,1,1)FOR K NOT EQUAL TO -1/2
THE NUMBER OF INDEPENDENT ELEMENTS IN THE BASIS IS CALLED THE DIMENSION.PLEASE NOTE THAT BASIS SHOULD CONSIST OF ONLY INDEPENDENT ELEMENTS.THERE ARE 3 ELEMENTS NAMELY.....(0,-1,K), (1,-1,0)AND (1,1,1)...HERE.SO THIS
HAS DIMENSION OF 3.
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