Write each vector as a linear combination of the vectors in

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 = (-2, -5, 5)

v =

(c)    w = (0, -15, 15)

w=

Solution

The set W of all linear combinations of s1, s2 are a subspace of S.

and W is the smallest subspace of S that contains s1, s2  . . . ,sr   for every other subspace

of S that contains s1, s2, . . . , sr must contain W

then have W is a subspace of S , it must be proven that it is closed

under addition and scalar multiplication. There is at least one vector in W,

namely, 0, since 0 = 0s1 + 0s2 If u and s are vectors in W, then

u = c1s1 + c2s2 +