euclidean n-space?

Question

find scalars c1,c2,c3 such that c1(1,1,0)+c2(1,0,1)+c3(0,1,1)=(1,-1,2)

c1(1,1,0)+c2(1,0,1)+c3(0,1,1)=(1,-1,2)
just wrote it again cause i dont know if its showing completely

equals (1,-1,2)
sorry

Answer 1

So
1) c1 + c2 = 1
2) c1 + c3 = -1
3) c2 + c3 = 2

from 1 and 2, c2 - c3 = 2, so c2 = 2, c3 = 0, c1 = -1

Answer 2

c1(1,1,0) + c2(1,0,1)+ c3(0,1,1) = (1,-1,2)
so
c1 + c2 =1
c1 + c3 = -1
c2+ c3 = 2
which is a system of linear equations
substract eq. 2 from eq1:
c2 - c3 = 2,
which together with eq. 3 means that
c3 = 0
so c2=2,
and
c1 = -1
.

Answer 3

What comes after the .....?

Those 3 vectors are linearly independent and span R^3.


equals (1,-1,2)

You're solving a system of 3 equations in 3 unknowns. I'd recommend writing them down is 'augmented matrix' form and using row reduction operations to solve.

Or invert the matrix

You've probably solved such equations, haven't you?