Question

Let S={(1,4),(0,3)} be a subset of R^2(R).prove that (2,3) belongs to L(S)

Answer 1

Let's v1 = (1,4) and v2 = (0,3). Now find a and b such that

(2,3) = a*v1 + b*v2

(2,3) = a*(1,4) + b*(0,3)

From here we have linear system

$\begin{cases} a*1 + b*0 = 2\\ a*4 + b*3 = 3 \end{cases}$

From first equation we find a = 2 and substitute it to second we find b = -5/3. Exist a, b in R such that (2,3) = a*v1 + b*v2, where v1, v2 in S

Hence (2,3) belongs to L(S) - the linear span of S.