秩定理

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.

Let $A$ be an $m\times n$ matrix, so the equation $Ax = 0$ has $n$ unknowns. Which identity does the rank theorem give?

A $3\times 6$ matrix $A$ has nullity $4$. Enter $\operatorname{rank}(A)$ as a number.