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# express vector as linear combination of 3 vectors

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1. 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 , 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

help?
2. 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.
3. hot damn that worked!
thanks a tonne
4. (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 , 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

help?
The linear combination of v1,v2,v3 is

where and v1,v2,v3 are independent linearly.
This vector equation means three scalar equations (for x, y, and z coordinates)

Solve simultaneously for

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