Given vectors u = (−1, 2, 3) and v = (3, 4, 2) in R 3 , consider the linear span: Span{u, v} := {αu + βv: α, β ∈ R}. Are the vectors (2, 6, 6) and (−9, −2, 5) in Span{u, v} ?

Step-by-step explanation:

Let . We have that if and only if we can find scalars such that . This can be translated to the following equations:

1.

2.

3.

Which is a system of 3 equations a 2 variables. We can take two of this equations, find the solutions for and check if the third equationd is fulfilled.

Case (2,6,6)

Using equations 1 and 2 we get

whose unique solutions are , but note that for this values, the third equation doesn't hold (3+2 = 5 6). So this vector is not in the generated space of u and v.

Case (-9,-2,5)

Using equations 1 and 2 we get

whose unique solutions are . Note that in this case, the third equation holds, since 3(3)+2(-2)=5. So this vector is in the generated space of u and v.

-6 is the correct option from the above given options