Question

1. Find a vector in R^3 which is not in the span of the set S-((1,2, -2), (2, -1, 1). Explain.

Answer 1

Explanation:

A vector in R^3 which is not in the span of the set S {(1,2,-2) and (2,-1,1)

If a vector is in the span it can be represented as a linear combination of these two vectors

Let S1 = (1,2,-2) and S2 = (2,-1,1)

i.e. any vector which is of the form

`\alpha S1+\beta S2\\=(\alpha +2\beta,2\alpha -\beta,-2\alpha +\beta)`

Where alpha and beta are any real numbers

Any vector not in this form will not be in the span

i.e. say if alpha = beta =1,

then spanned vector = (3,1,-1)

If we change one coordinate alone say

(3,0,-1) this cannot be represented as a linear combination hence this would be the answer.