Question

What is the relationship between linear independance and filling up the whole space?

Answer 1

What is your exact question?
If we use 3-dim as an example,
the x-y plane can be described by two vectors (e.g. the x-axis and the y-axis)
There are many other vectors we could use , any two "directions" that lie in the plane.
One vector would not be enough to "reach" all points in the plane. Two independent vectors would be. More than two would allow you to reach every point in the plane, but more than two vectors would be redundant.
And no matter how many vectors you have that all lie in the x-y plane, you will not be able to reach a point outside that plane, without a vector that is independent of all the vectors in the x-y plane.

Answer 2

Filling up the whole space needs enough vectors which are linear independent .
Filling up a Vector Space of n dimensions needs at least n linear independent vectors.

Answer 3

Let us suppose there vector A=[1 2 -1], B=[2 3 0], C=[-1 2 5]. If these three vectors are linearly independent (which actually is) then they can be reduced to identity matrix of order 3 i.e. I\[I{3}\].By the linear combination of three vectors of I3 any vector in R3 can be created. Hence we can say that linear combination of I3(Identity matrix of order 3) filled R3(vector space).In other words linear combination of A, B, C fill entire R3.