Theorem

If two matrices $A$ and $B$ are row equivalent, then their row spaces are the same. If $B$ is in echelon form, the nonzero rows of $B$ form a basis for the row space of $A$ as well as for that of $B$.

The Rank Theorem

The dimensions of the column space and the row space of an $m \times n$ matrix $A$ are equal. This common dimension, the rank of $A$, also equals the number of pivot positions in $A$ and satisfies the equation

$$\text{rank } A + \dim \text{Nul } A = n$$

The Invertible Matrix Theorem (Continued)

Let $A$ be an $n \times n$ matrix. Then the following statements are each equivalent to the statement that $A$ is an invertible matrix:

m. The columns of $A$ form a basis of $\mathbb{R}^n$.
n. $\text{Col } A = \mathbb{R}^n$.
o. $\dim \text{Col } A = n$.
p. $\text{rank } A = n$.
q. $\text{Nul } A = \{ \vec{0} \}$.
r. $\dim \text{Nul } A = 0$.