Q1: Let $\ell: V \rightarrow W$ be a linear map. What is correct?

A1: $\mathrm{Ran}(\ell) \subseteq W$

A2: $\mathrm{Ran}(\ell) \subseteq V$

A3: $\mathrm{Ran}(\ell) = V$

A4: $\mathrm{Ran}(\ell) = W$

A5: $\mathrm{Ran}(\ell) \in V$

A6: $\mathrm{Ran}(\ell) \in W$

Q2: Let $\ell: V \rightarrow W$ be a linear map. What is correct?

A1: $\mathrm{Ker}(\ell) \subseteq V$

A2: $\mathrm{Ker}(\ell) \subseteq W$

A3: $\mathrm{Ker}(\ell) = V$

A4: $\mathrm{Ker}(\ell) = W$

A5: $\mathrm{Ker}(\ell) \in V$

A6: $\mathrm{Ker}(\ell) \in W$

Q3: Let $\ell: V \rightarrow W$ be a linear map where $\mathrm{dim}(V) = 5$. What is never possible?

A1: $\mathrm{Ker}(\ell)$ is 4-dimensional and $\mathrm{Ran}(\ell)$ is 2-dimensional.

A2: $\mathrm{Ker}(\ell)$ is 2-dimensional and $\mathrm{Ran}(\ell)$ is 3-dimensional.

A3: $\mathrm{Ker}(\ell)$ is 5-dimensional.

A4: $\mathrm{Ran}(\ell)$ is 5-dimensional.