Write each vector as a linear combination of the vectors in S. (If not possible, enter IMPOSSIBLE.) S = {(6, −7, 8, 6), (4, 6, −4, 1)} (a) u = (2, 19, −16, −4) u = −1 s1 + 2 s2 (b) v = 43 2 , 113 4 , −18, 13 2

a) Yes, it is a linear combination.

b) Impossible to write as a linear combination.

Step-by-step explanation:

Recall that given vectors u,v,w we say that w is a linear combination of u and v if there exists real numbers a,b such that

a) . So, we have the following

. Which give us the following equations

Note that if we add the first and the third equation, we get that which implies that a=-1. In the first equation, if a=-1, then which implies that b=2. We must check that when (a,b) =(-1,2) the four equations are still valid.

So

Since all equations are met, we have written the desired vector u as the linear combination of the initial vectors.

b) Repeating the same analysis, we get

adding the first and third equation we get so a = 207/7. Replacing this value will give us that b=891/14.

However,

note that

. Then, it is impossible to write the linear combination.

set each factor equal to 0

4x-3 = 0

2x-1 = 0

solve for x.

4x-3=0

4x=3

x= 3/4

solve for x. 

2x-1= 0

2x= 1

x= 1/2

eventually after a few more steps, your answer should be 1/2 is less than or equal to x and 3/4 is greater than or equal to x