This is more of a conceptual question. Here’s what I know about a linearly independent set of vectors:
A set of vectors \{v_1, \ldots, v_p\} is linearly independent if the equation
x_1v_1 + x_2v_2 + \ldots + x_pv_p = 0
has only the solution x = 0, the trivial solution.
Here’s what I know about spans:
\operatorname{span} \{v_1, \ldots, v_p\} is the set of all linear combinations of v_1, \ldots, v_p.
Given a set of vectors S = \{v_1, v_2, \ldots, v_k\} in a vector space V, S is said to span V
if \operatorname{span}(S) = V.
The columns of a matrix A span \Bbb{R}^m if and only if A has a pivot position in every row.
Also, a matrix A spans \Bbb{R}^m if the number of nonzero rows when in reduced row echelon form equals m.
My confusion is that these two things —being linearly independent and spanning \Bbb{R}^m— seem like the same thing. For example, if a matrix A has a pivot position in every row, doesn’t that mean A is also linearly independent? The only example I can think of that proves they are not the same is if a given matrix has no free variables (linearly independent) but has some zero rows and so does not span \Bbb{R}^m.
I am just trying to really grasp the concepts here since it seems they correlate to many different topics in linear algebra. Any further explanation would be much appreciated! Thanks!
