Say there are two independent vectors in R3, to be specific we can assume the vectors to be [1 0 0] & [0 1 0]. I'm confused what will be the span of these vector and what will be its dimension? actually my doubt arises from the link https://www.youtube.com/watch?v=AqXOYgpbMBM in this video there are 4 vectors in R4 but only 3 of them are linearly independent therefore it spans R3 and not R4. So my question is, is it possible to span lower dimension when there are higher number of components in a vector? I'm not able to visualize it Please help!
Yes, of course. A plane has two dimensions even if the coordinate are three. In a cartesian plane a straight line has 1 dimension but its points have 2 coordinates. A point alone has zero dimension whatever is the dimension of the space it is in.
Hope this helps
I think you misunderstood the assertion. A set of vectors in $\mathbf R^n$ cannot possibly generate $\mathbf R^p$ if $p < n$, ut it can generate a subspace of dimension $p$, i.e. a subspace isomorphic to $\mathbf R^p$.