Q1: What is the correct definition for the rank of a matrix $A \in \mathbb{R}^{m\times n}$?

A1: $\mathrm{rank}(A) = \mathrm{dim}\Big({ A \mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n } \Big) $

A2: $\mathrm{rank}(A) = \mathrm{dim}\Big( { \mathbf{x} \mid A \mathbf{x} \in \mathbb{R}^m } \Big)$

A3: $\mathrm{rank}(A) = \mathrm{dim} \Big( { \mathbf{y} \mid \mathbf{y} \in \mathbb{R}^m } \Big)$

A4: $\mathrm{rank}(A) = \mathrm{dim} \Big( { A^T \mathbf{x} \mid A \mathbf{x} \in \mathbb{R}^n } \Big)$

A5: $\mathrm{rank}(A) = \mathrm{dim} \Big( { \mathbf{x} \in \mathbb{R}^n \mid A \mathbf{x} = 0 }\Big)$

Q2: What is the correct formulation of the rank-nullity theorem for a matrix $A \in \mathbb{R}^{m\times n}$?

A1: $\mathrm{dim} \Big( \mathrm{Ker}(A) \Big) + \mathrm{dim} \Big( \mathrm{Ran}(A) \Big) = n$

A2: $\mathrm{dim} \Big( \mathrm{Ker}(A) \Big) + \mathrm{dim} \Big( \mathrm{Ran}(A) \Big) = m$

A3: $\mathrm{dim} \Big( \mathrm{Ker}(A) \Big) + \mathrm{dim} \Big( \mathrm{Ran}(A) \Big) + n = 0$

A4: $\mathrm{dim} \Big( \mathrm{Ker}(A) \Big) + \mathrm{dim} \Big( \mathrm{Ran}(A) \Big) = n-m$

Q3: What is the rank of the matrix $\begin{pmatrix} 2 & 1 \ 0 & 0 \end{pmatrix}$?

A1: $1$

A2: $0$

A3: $2$

A4: $3$

Q4: Let $A \in \mathbb{R}^{4 \times 3}$ be matrix with rank $1$. What is the dimension of the kernel of $A$?

A1: $2$

A2: $0$

A3: $1$

A4: $3$