Q:

How to tell if a set S, spans a plane, a line, or a point? 

Okay, I realize this question has been asked a lot, but I am having trouble with it. 
Look at this question for instance :
Determine whether the set $S$ spans $\Bbb R^3$. If the set does not span $\Bbb R^3$, then give a geometric description of the subspace that it does span. 
$$S = \{(2, 0, 1), (2, 0, −1), (6, 0, 1), (4, 0, 2)\}$$
Answer: $S$ does not span $\Bbb R^3$. $S$ spans a plane in $\Bbb R^3$
Now, I know that the determinant is Zero, therefore it doesn't span $\Bbb R^3$, but how do I know if it spans a plane, a line or a point? 
Thank you in advance. 

A:

A single non-zero vector spans a line. 
If two vectors $a,b$ are linear independent (both vectors non-zero and there is no real number $t$ with $a=bt$), they span a plane. 
To span $\mathbb R_3$, you need $3$ linear independent vectors. 
In general, vectors $a_1,a_2,\cdots,a_n$ are linear independent , if $t_1a_1+t_2a_2+\cdots t_na_n=0$ implies $t_1=t_2=\cdots t_n=0$. In this case, they span $\mathbb R_n$

