Write each vector as a linear combination of the vectors in

Write each vector as a linear combination of the vectors in S. (Use s_1 and s_2, respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.) S = {(1,2, -2), (2, -1, 1)} (a) z = (-8, -1,1) z = (b) v = (-1,-6, 6) v = (c) w = (-1, -17, 17) w = (d) u = (2, -6, -6) u =

Solution

Let A = [s₁,s₂,z,v,w,u]=

1

2

-8

-1

-1

2

2

-1

-1

-6

-17

-6

-2

1

1

6

17

-6

In order to express z,v,w,u as linear combinations of s₁,s_2, we will reduce A to its RREF as under:

Add -2 times the 1st row to the 2nd row

Add 2 times the 1st row to the 3rd row

Multiply the 2nd row by -1/5           

Add -5 times the 2nd row to the 3rd row

Multiply the 3rd row by -1/12          

Add -2 times the 3rd row to the 2nd row

Add -2 times the 3rd row to the 1st row

Add -2 times the 2nd row to the 1st row

Then the RREF of A is

1

0

-2

-13/5

-7

0

0

1

-3

4/5

3

0

0

0

0

0

0

1

Now, it is apparent that z = -2s₁ -3s₂, v = (-13/5)s₁ +(4/5)s₂, and w = -7s₁+3s₂. However, u cannot be expressed as a linear combination of s₁,s₂.

1	2	-8	-1	-1	2
2	-1	-1	-6	-17	-6
-2	1	1	6	17	-6