Question

Find an element f(x) of the span of {sin x, 1} in the vector space F(R) such that f(x)>0 for all x

Answer 1

If the vector w is in the span of 2 vectors v1 and v2 then we can express w as a linear combination of v1 and v2.
\[\mathbf{w} = c_1 \mathbf{v}_1 + c_2\mathbf{v}_2\]So in this case you are looking for a linear combination of sin(x) and 1 which meets the requirement.