How is it proved (intuitively) that the more independent vectors the "larger" the vector subspace will be

Question

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!

Answer 1

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.