Given $v1,v2,\ldots,vk,u,w$ in a vectorial space $V$, it is said that $x_1v_1+...+x_kv_k=u$ has a single solution and that $x_1v_1+...+x_kv_k=w$ has no solution.
How do you find the dimension of $\mathrm{Sp}(v_1,...,v_k,w)$?
From what I understand the latter says that $w$ is independent of $v_1,...,v_k$ so it can span, but I don't understand how to prove it and what is the connection of vector $u$.
Thank you very much for your help.
Saying that for some vector $u$, the linear system $x_1v_1+\dots x_k x_k=u$ has a unique solution means: 1) $u$ lies in the span $\langle v_1, \dots, v_k\rangle$ (existence of a solution) and 2) the vectors are linearly independent (uniqueness). Therefore they generate a subspace of dimension $k$.
On the other hand, saying that the linear system $x_1v_1+\dots x_k x_k=w$ has no solution means, as you noticed, $w$ is independent of $v_1,\dots, v_k$, so that the system of vectors $v_1,\dots, v_k,w$ is linearly independent, and thus generates a subspace of dimension $k+1$.