The span of two parallel line segments I'm trying to understand how to think about the span of two parallel line segments.
Let $\Pi_0$ be the vector space of all line segments through the origin. Then, the span of two line segments through the origin $\vec{o}$ is, $${span}(\vec{x}, \vec{y})= a\vec{x}+b\vec{y} = \Pi_0.$$
However, what happens when both vectors do not intersect at $\vec{o}$ or are parallel. I understand that this is not a subspace of $\Pi_0$ but, is it even defined, or can it be a vector space and/or a subspace of another vector space. How does a linear combination of these two vectors look like geometrically?
Your feedback will be much appreciated.
 A: Perhaps to answer your question, you need to remember a very useful theorem in linear algebra. This theorem states that:

Theorem: Let $(V,\oplus, \odot, (F,+,\cdot))$ a vector space over a field $F$ with the operations $\oplus$ and $\odot$ sums, and scalar-vector product, respectively. Let $H\subseteq V$. We will say that $H$ is a vector subspace if, and only if,

*

*$H\not=\emptyset \iff \vec{0}_{V}\in H$.

*$\forall h_{1},h_{2} \in H: h_{1}\oplus h_{2} \in H$.

*$\forall \alpha \in F, \forall h \in H: \alpha \odot \in H$.


Now, I have placed emphasis on identifying the sum and product operation, because here is the key to the answer to one of your questions. I will try to answer your questions one by one.
For example, I will explain this using the vector space $\mathbb{R}^{2}$.

*

*Indeed, let the vector space of $\mathbb{R}^{2}$ and let $\Pi_{0}\subseteq \mathbb{R}^{2}$ with de conventional operations so $$\Pi_{0}=\left\{\begin{pmatrix} x \\ y \end{pmatrix} \in \mathbb{R}^{2}: ax+by=0, a,b \in \mathbb{R} \right\}=\text{span} \left\{ \begin{pmatrix} -b \\ a \end{pmatrix}\right\}$$
is a vector subspace of $\mathbb{R}^{2}$


*If the vectors do not intersect in $(0,0)$ that means that the zero vector of $\mathbb{R}^{2}$ isn't in $\Pi_{0}$ for example and that contradicts our first property of the theorem, therefore it would not be a vector subspace.


*The reason I already mentioned and emphasized the notation associated with operations in vector space is the key to the answer to your question.
You can define a vector space with unconventional operations where you can change the geometry of vector spaces where some "weird" things happen.


*Take for example $\mathbb{R}^{2}$ is a vector space rich in geometry and surely you will be able to observe how the linear combination of the vectors that are subject to the condition that I indicated in $\Pi_{0}$.
