The formal definition of basis is:
A basis of a vector space $V$ is defined as a subset $v_1, v_2, . . . , v_n$ of vectors in that are linearly independent and span vector space $V$.
The definition of spanning is:
A set of vectors spans a space if their linear combinations fill the space.
But what is the intuitive meaning of this, and the idea of a vector span? All I know how to do is the process of solving by putting a matrix into reduced row-echelon form.
Separately, I"m not sure if I should put this in a new question, but could someone relate this to an intuitive explanation for the row space and column space? So a column space is all the linear combinations of each column of matrix $A$. So what? What does this imply? And a row space is, is it a linear combination of all the rows of $A$, because the book just says its the column space of $A^T$, which I hope means the same thing. So, sure, that's what the definitions of the row space and column space are, but how do all these concepts relate? I'm getting especially confused getting to the fundamental theorems of linear algebra part where we talk about row space, column space, and nullspaces all together.