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

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

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

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

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

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

Q2: What is the correct definition for the kernel of a matrix $A \in \mathbb{R}^{m\times n}$?

A1: $\mathrm{Ker}(A) = { \mathbf{x} \in \mathbb{R}^n \mid A \mathbf{x} = 0 }$

A2: $\mathrm{Ker}(A) = { \mathbf{x} \in \mathbb{R}^n \mid A \mathbf{x} = \mathbf{y} }$

A3: $\mathrm{Ker}(A) = { A^{-1} \mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n }$

A4: $\mathrm{Ker}(A) = { A \mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n }$

Q3: What is not the range of the matrix $\begin{pmatrix} 2 & 1 \ 0 & 0 \end{pmatrix}$

A1: $\mathrm{Ran}(A) = \mathrm{Span}( \begin{pmatrix} 2 \ 1 \end{pmatrix})$

A2: $\mathrm{Ran}(A) = \mathrm{Span}( \mathbf{e}_1 )$

A3: $\mathrm{Ran}(A) = \mathrm{Span}( \begin{pmatrix} 1 \ 0 \end{pmatrix} )$

A4: $\mathrm{Ran}(A) = \mathrm{Span}(\begin{pmatrix} 2 \ 0 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \end{pmatrix} )$