Assume we have two vectors with five components.
Namely, $v = (v_1, v_2, v_3, v_4, v_5)$ and $u = (u_1, u_2, u_3, u_4, u_5)$.
I know that if they are linearly independent, the space that they span has dimension $2$ and I know that these vectors are not unique for that space (can be spanned by some other two vectors).
Question is, how can we tell if given another two vectors spans the same space?
Say, $x = (x_1, x_2, x_3, x_4, x_5)$ and $y = (y_1, y_2, y_3, y_4, y_5)$.
Are linear combinations of $u$ and $v$ equal to linear combinations of $x$ and $y$? How can we answer such a question for the given vectors?
In other words, can we find a $(c, d)$ for every $(a, b)$ such that $au + bv = cx + dy$?
Best Answer
If $x,y \in span(v,u)$ and if $x,y$ are linearly independent in $span(v,u)$ then $span(v,u)=span(x,y)$. This is due to the fact that if x and y are in the span of v and u then the span of x and y must be in it. Since they are both linearly independent this implies that $dim span(x,y) = dim span (u,v) $ implying that x,y is a basis of $span(u,v)$.