Why $L$ is a line? My professor wrote:
Let $\begin{pmatrix}
        a\\
        b\\
        c
    \end{pmatrix}\in \mathbb{R}^3$ then $L = \operatorname{Span}\{\begin{pmatrix}
        a\\
        b\\
        c
    \end{pmatrix}\} =$ line.
My question is: Why $\operatorname{Span}\{\begin{pmatrix}
        a\\
        b\\
        c
    \end{pmatrix}\}$ is a line? could someone explain this to me please? My background is that I know the slope-intercept form, two-points form and the point-slope form equations of the line but still I can not see why the above column vector spans a line.
Thanks!
 A: You can try and google the definition of a line in the $3D$ space .
First of all what is $\text{span}(\{(a,b,c)^{T}\}$ ? .
As a set it is precisely $\{t(a,b,c)\,:t\in\mathbb{R}\}$.
Now compare this to the case of $\mathbb{R}^{2}$. Say you had $(a,b)$ instead. Then the span would be $\{t(a,b)=(ta,tb):t\in\mathbb{R}\}$.
Compare this with the technique you actually use to draw a line with a ruler and pencil. Once you are given a point and a direction , or two points, you join them using the ruler and extend it on both sides. Isn't it analogous to considering a point of starting point $O$ with it's position vector given by $\vec{p}$ and a direction given by $\vec{v}$ and just saying that all the points in your line are the same as those with coordinates $\vec{p}+t\vec{v}\,\,$ for all $-\infty<t<\infty$? .
The idea is precisely the same for higher dimensions.
A line in a higher dimension passing through two points $\vec{p}$ and $\vec{q}$ is as a set precisely equal to $\{\vec{p}+t(\vec{q}-\vec{p}):t\in\mathbb{R}\}$.
This is why they are saying that it is a line.
You can also see my answer here
