Why a inconsistent system can be considered as a span? (updated, example added) I have some problems in learning about span.
The definition of $\operatorname{span}\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}=$ all linear combinations of $\vec{v}_1,\vec{v}_2,\vec{v}_3$.
However, when I deal with question like asking a given vector where it is inside the span of given vector $\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}$. I try to solve it as a system $\begin{bmatrix}\vec{v}_1&amp;\vec{v}_2&amp;\vec{v}_3\end{bmatrix}\cdot\begin{bmatrix}t_1&amp;t_2&amp;t_3\end{bmatrix}^T=v$.
I found the system is inconsistent, but $v$ is still an element of $\operatorname{span}\{\vec{v}_1,\vec{v}_2,\vec{v_3}\}$, why?
an example:
If $v$ is in the span of the set you mentioned, then it is expressible as a linear combination of vectors in the original set. This linear combination solves the equation you say is "inconsistent".

If what you mean by "inconsistent" is that a matrix made of the $v_i$ has zero determinant, this simply expresses the fact that they are linearly dependent, and that they will span a space of fewer than $3$ dimensions. But the span still exists. If $v_1\neq 0$ then $v_1$ is definitely in the span, for example.
the only v is in the span {v1, v2, v3} if the system you mentioned is consistent. Not sure if your last sentence is true.