Question

Write each vector as a linear combination of the vectors in S. (Use s1​ and s2​, respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.) s={(1,2,−2),(2,−1,1)}

(a) z=(−7,1,−1) z= ________________
(b) v=(−1,−5,5) v= ________________
(c) w=(4,−17,17) w= _______________
(d) u=(3,−6,−6) u= ______________

Answer 1

To write each vector as a linear combination of vectors in S: (a) z=(-7,1,-1) is -4s1 + s2, (b) v=(-1,-5,5) is impossible, (c) w=(4,-17,17) is 5s1 + 2s2, and (d) u=(3,-6,-6) is impossible.

Step-by-step explanation:

To write each vector as a linear combination of the vectors in S, we need to find scalar values such that multiplying each vector in S by its corresponding scalar and adding them together will give us the given vector. Let's calculate:

(a) For vector z=(-7,1,-1), we need to find scalars x and y such that x(1,2,-2) + y(2,-1,1) = (-7,1,-1). Solving the system of equations, we find x = -4 and y = 1. Therefore, z = -4s1 + s2.

(b) For vector v=(-1,-5,5), there are no scalar values that can be multiplied with the vectors in S to get v. Therefore, it is impossible to write v as a linear combination of the vectors in S.

(c) For vector w=(4,-17,17), we need to find scalar values x and y such that x(1,2,-2) + y(2,-1,1) = (4,-17,17). Solving the system of equations, we find x = 5 and y = 2. Therefore, w = 5s1 + 2s2.

(d) For vector u=(3,-6,-6), there are no scalar values that can be multiplied with the vectors in S to get u. Therefore, it is IMPOSSIBLE to write u as a linear combination of the vectors in S.