Let's say we have the vectors $(3,2), (4,2), (4,4), (3,3), (3,4), (4,3), (2,4), (2,3)$. Is there any way to show that any vector in the form $(n,m)$ where $n,m \in \{ 2, 3, ..., 9, 10 \}$can be formed by the aforementioned set of vectors?
I realize that this is like solving a system of equations in two variables, but that leads to $8$ variables and $2$ equations... I then thought that maybe we could take the determinant of the coefficient matrix, but that clearly doesn't work because the matrix isn't a square one.
Could someone please help?
If you want to show that you can form a 2-dimensional vector, then you just need to show that two of the vectors you listed are linearly independent, i.e. show that $x_1 (a_1, a_2) + x_2 (b_1, b_2) = (0,0)$ has a solution only when both $x_1$ and $x_2$ is equal to zero.