The rank of a matrix is the number of non-zero rows in its RREF.
Theorem: Injective means Tx = 0 is trivial
Theorem: Ax = 0 is trivial means columns of A are linearly independent
Theorem: If the columns of A are L.I., then the number of pivots equals n.
Theorem: A is surjective means that there m pivots.
We conclude that A has m pivots.
A row space is the span of the rows of a matrix. The row space of two row-equivalent matrices is the same. Why? because to row reduce you just add linear combinations of rows. The basis of the row space is equal to the non-zero rows in the row-reduced matrix, which equals the dimension of the row space.
The column space is the span of the columns of A. In other words, it’s the image of a matrix A. I will prove that the dimension of the column space equals the dimension of the row space which equals the rank.
The null space is the set of all x such that Ax = 0. I will later prove that the Rank + Dim (Null Space) = n (columns in A).