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express vector as linear combination of 3 vectors

Express the vector (a,b,c,) as a linear combination of v1, v2 and v3 with coefficients in terms of the contstant a,b and c
v1= (1,-1,0)
v2= (1,0,-1)
v3= (1,1,1)

absolutely no idea where to start :frown: , never been taught this and it isn't covered in any textbook i have. looked at the answer in the back for a clue and it says:
(1/3)(a-2b+c)v1 + (1/3)(a+b-2c)v2 + (1/3)(a+b+c)v3

which has confused me even more as i can't see how i'd get to that at all :frown:

help?
if it were me doing this question, I would first try and create the vectors (1,0,0) (0,1,0) and (0,0,1) from v1 v2 and v3, then the answer should be obvious.

so (v1+v2+v3)/3=(1,0,0) so then a(v1+v2+v3)/3=(a,0,0)

Doing a same thing for the others yields the result. I hope this was clear.
Reply 2
hot damn that worked!
thanks a tonne
Reply 3
Original post by Misiak
Express the vector (a,b,c,) as a linear combination of v1, v2 and v3 with coefficients in terms of the contstant a,b and c
v1= (1,-1,0)
v2= (1,0,-1)
v3= (1,1,1)

absolutely no idea where to start :frown: , never been taught this and it isn't covered in any textbook i have. looked at the answer in the back for a clue and it says:
(1/3)(a-2b+c)v1 + (1/3)(a+b-2c)v2 + (1/3)(a+b+c)v3

which has confused me even more as i can't see how i'd get to that at all :frown:

help?


The linear combination of v1,v2,v3 is
λ1v1+λ2v2+λ3v3=v\displaystyle \lambda_1 \vec v_1 +\lambda_2 \vec v_2 +\lambda_3 \vec v_3 = \vec v
where λ1,λ2,λ3R\displaystyle \lambda_1, \lambda_2, \lambda_3 \in R and v1,v2,v3 are independent linearly.
This vector equation means three scalar equations (for x, y, and z coordinates)
λ11+λ21+λ31=a\lambda_1\cdot 1 +\lambda_2 \cdot 1+\lambda_3 \cdot 1=a
λ1(1)+λ31=b\lambda_1\cdot (-1) +\lambda_3 \cdot 1=b
λ2(1)+λ31=c\lambda_2 \cdot (-1)+\lambda_3 \cdot 1=c
Solve simultaneously for λi\lambda_i
(edited 13 years ago)

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