Dimension of the span of two parallel lines in $R^4$.

Question

I am asked if the following question is true or false:

Let $r,s$ be two parallel lines in $R^4$ then the dimension of $Span(r \cup s)$ is strictly less than $3$.

I think this is true because two parallel lines in $R^4$ can be described by the following two systems

\begin{cases} x = x_0 + at \\ y = y_0 + bt \\ z = z_0 + ct \\ w = w_0 + dt \end{cases}

\begin{cases} x = x_1 + at \\ y = y_1 + bt \\ z = z_1 + ct \\ w = w_1 + dt \end{cases}

and so the max dimension the span can obtain is when the vectors \begin{pmatrix} x_0 \\ y_0 \\ z_0 \\ w_0 \end{pmatrix} \begin{pmatrix} x_1 \\ y_1 \\ z_1 \\ w_1 \end{pmatrix}

are linearly independent, and because they are $2$ the max dimension obtained is $2$.

In the solutions the answer is instead false, why?

Answer 1

In fact the span of those two lines is the span of the three vectors $(x_0,y_0,z_0,w_0),(x_1,y_1,z_1,w_1),(a,b,c,d)$.

Answer 2

The two vectors you've described are each a single point on one of the two lines; by taking their span, you get a space of dimension at most 2, but you're not taking the span of the entire lines.

Geometrically, if you imagine a single line that does not pass through the origin, and you take its span, you get a 2D plane (imagine a triangle where one vertex is the origin, and extend the line segment opposite the origin to a line). You already have a 2D space with just one line; if you take another line, but lift it from this 2D plane, adding it and taking the span will increase the dimension.