Question 26495: I have a homework problem that I don't know how to do.
I'm asked to solve the following problem;
For what values of h is v3 in span {v1,v2} and for what values of h is {v1,v2,v3}linearly dependent?
given that
----[ 1]-----[-2]------[ 2]
v1=[-5]-v2=[10]--v3=[-9]
----[-3]-----[ 6]-------[ h]
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! I have a homework problem that I don't know how to do.
I'm asked to solve the following problem;
For what values of h is v3 in span {v1,v2} and for what values of h is {v1,v2,v3}linearly dependent?
given that
----[ 1]-----[-2]------[ 2]
v1=[-5]-v2=[10]--v3=[-9]
----[-3]-----[ 6]-------[ h]
LET US TAKE 3D SPACE WITH,I,J AND K AS UNIT VECTORS...SO
V1=1I-5J-3K
V2=-2I+10J+6K
V3=2I-9J+HK
WE FIND HERE THAT V2=-2V1...SO THEY ARE NOT INDEPENDENT...PLEASE CHECK THE NUMBERS AND COME BACK.HOWEVER ,I AM GIVING THE METHOD OF SOLUTION BELOW..
IF V3 IS IN V1 AND V2 SPAN ,THEN WE HAVE
V3=AV1+BV2=
2I-9J+HK=AI-5AJ-3AK-2BI+10BJ+6BK= I(A-2B)+J(10B-5A)+K(6B-3A)
HENCE SOLVE FOR A AND B FROM EQNS.I AND II AND SUBSTITUTE THOSE VALUES IN EQN.III TO GET H VALUE..HERE YOU WILL NOT GET THE ANSWER.
A-2B=2.......................I
10B-5A=-9....................II
6B-3A=H.......................III
THE ABOVE CASE IS FOR V1,V2,V3 TO BE DEPENDENT.
SO CHECK YOUR SECOND QUESTION AGAIN..THEY ARE INDEPENDENT OR DEPENDENT?
IF V1,V2,V3 ARE TO BE INDEPENDENT ,H SHOULD NOT BE EQUAL TO THE VALUE OBTAINED AS ABOVE.
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