Question 30580: I want to find about(three things)the set, W={(1,0,0),(0,1,0),(0,0,1),(-7,8,-9)}
Is this linearly dependent? Does this spans R3? Is this a basis for R3?
I think this set spansR3 because the first three vectors span R3, so adding the last one does not change it. Also, I think it is linearly dependent as there are nontrivial solutions. But, I am not sure if it is not a vasis for R3. Could you give me advice???
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! I want to find about(three things)the set, W={A=(1,0,0),B=(0,1,0),C=(0,0,1),D=(-7,8,-9)}
Is this linearly dependent?
YES FOR REASON GIVEN BY YOU ..WE CAN SHOW AS FOLLOWS EASILY
WE NOTICE THAT
D=-7A+8B-9C ...OR
WE ARE ABLE TO FIND
PA+QB+RC+SD=0...(-7A+8B-9C-D=0),WHILE
P,Q,R,S ARE NOT ALL EQUAL TO ZERO.HENCE THEY ARE LINEARLY DEPENDENT.
Does this spans R3?
YES
Is this a basis for R3?
FOR THIS YOU HAVE TO REFER DEFINITION OF BASIS..IT STIPULATES..W A SUBSET OF VECTOR SPACE R3 IS A BASIS FOR R3 IF
1.W CONSISTS OF LINEARLY INDEPENDENT VECTORS
2.W GENERATES R3 OR EACH VECTOR IN R3 IS A LINEAR COMBINATION OF FINITE NUMBER OF ELEMENTS OF W .
SO WE HAVE CONDITION 2.SATISFIED ..BUT NOT CONDITION 1..SO W IS NOT A BASIS.
I think this set spansR3 because the first three vectors span R3,
VERY GOOD
so adding the last one does not change it.
CORRECT
Also, I think it is linearly dependent as there are nontrivial solutions.
PERFECT AS WE SAW ABOVE
But, I am not sure if it is not a vasis for R3. Could you give me advice???
YOU HAVE DONE EXTREMELY WELL...KEEP IT UP..YOU JUST NEED A LITTLE POLISHING...GOOD LUCK.
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