Confused on linear combination

How to determine whether the given vector "u" can be written as a linear combination of the vector in "s"

s= {(1,2,-2),(2,-1,1)}
u= {(-4,-3,3)}

All the vectors that can be written as a linear combination of the vectors in "s" lie in a plane defined by the vectors in "s". So, form the vector product of the vectors in "s" to get a vector perpendicular to this plane. Then the scalar product of this new vector with any vector lying in the plane will be zero. This can be used as a test of whether "u" lies in that plane, and therefore of whether it can be written as a linear combination of the vectors in "s"

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The vector product of (1,2,-2) and (2,-1,1) is (0, -5, -5). And the scalar product of this with (-4,-3,3) is

0x-4 + -3x-5 + 3x-5 = 0

Therefore (-4,-3,3) lies in the plane and it can be written as a linear combination of (1,2,-2) and (2,-1,1)

Yes you're on the right track. When the question asks whether u can be written as linear combination of the vectors in S, it's asking if you can find c1 and c2 such that

c1(1,2,-2)+c2(2,-1,1)= (-4, -3, 3) (which is what you wrote exactly)

You converted this to a system of linear equations. Now set up an augmented matrix and solve that.

Key point: notice the vectors on the left hand side of the augmented bar are the vectors from S exactly and that the vector on the right hand side of the augmented bar is the vector u. Whenever you want to see whether you can write a vector as a linear combination of a set of vectors, you can always do this by setting up the augmented matrix as above and seeing if it has a solution.