Ok, so from what I understand, if you have the vectors v1, v2....vk and you write them in column form to get the augmented matrix (A|0), they are linearly dependent if rank(A) < k and independent if rank(A) = k. If you write them in row form, as
v1
v2
.
.
vk
Then they are linearly dependent if there is one or more zero rows rank< m. Correct me if any of the previous stuff is incorrect. But then the book says the set of m vectors in R^n is linearly dependent if m > n. Can someone explain this? And also what's the difference between spans and spanning sets and how do they relate to linear dependence? Thanks.
Spanning set is to be thought of some kind of seeds. What these seeds yield is their span. Here yield means vectors obtained by the process of linear combination of a given set (the seed vectors
Linear independence is about how economical one can be with set of vectors if the aim is to consider all possible linear combinations. Suppose one vector $u$ is actually a linear combination of $v_1,v_,\ldots, v_n$ (that is $u$ is in the span of $v_i$'s). Then the larger set $\{u, v_1,v_2,\ldots, v_n\}$ does not contain any new vector in the span as the span of the set without $u$.
A set of vectors is linearly independent if none among them is in the span of the rest of the vectors. Linear independence will ensure there is no redundancy.