The Rank Theorem

Introduction

A matrix $A$ has $3$ rows, $5$ columns and rank $3$. Which of the following is the nullity of $A$?

A matrix $A$ has $4$ rows, $3$ columns and rank $3$. How many free variables does the equation $Ax=0$ have?

Complete the sentence: the dimension of the column space of a matrix $A$ is called the ______ of $A$. Enter one word.

A matrix $A$ has $4$ columns and nullity $1$. What is $\operatorname{rank}(A)$? Enter a number.

A matrix $A$ has $5$ rows, $4$ columns and rank $3$. Enter the dimension of the left null space of $A$, that is, the null space of $A^T$. Enter a number.

In your own words, state the single most important definition or theorem of this section, and explain what it says.