TITLE: How is it proved (intuitively) that the more independent vectors the "larger" the vector subspace will be
QUESTION [0 upvotes]: Say there is just one vector:
$$\begin{bmatrix} 1 \\ 2 \\ 3\end{bmatrix}$$
That vector,can "make up"/form just a line.
Now,if one more (independent) vector is appended then:
$$\begin{bmatrix} 1 & 0 \\ 
                  2 & 9 \\
                  3 & 10 \end{bmatrix}$$ 
These two vectors,can form a plane.
And at last,if one more vector is added:
$$\begin{bmatrix} 1 & 0 & 0 \\ 
                  2 & 9 & 0 \\
                  3 & 10 & 1 \end{bmatrix}$$
then,
$$ \mathbb{R^3}$$
is produced
Why is that?A fairly simpler question would be: why the sum of two vectors laying on the same plane result in a vector also in that plane?
Thanks!

REPLY [0 votes]: Because a plane (passing through the origin) is a vector subspace and, by definition, the sum of two elements of a subspace also belongs to that subspace.
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