unique 2-subspace of the set ${{\cal A}}$.

Representation of Subspaces {#sec:represent}
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The construction of the derived $q$-Fano plane and the residual $q$-Fano plane will be presented in Section \[sec:construction\]. The construction will start with subspaces from ${{\mathbb{F}}}_q^4$ which will consists of the unique 0-subspace of ${{\mathbb{F}}}_q^4$ and the subspaces of the sets ${{\cal A}}$, ${{\cal B}}$, ${{\cal C}}$, and ${{\cal D}}$. These subspaces will be extended and/or expanded to 2-subspaces in ${{\mathbb{F}}}_q^6$ for the derived $q$-Fano plane, and to 3-subspaces in ${{\mathbb{F}}}_q^6$ for the residual $q$-Fano plane. Most of these extensions will be performed with the extension space and hence the representations of these subspaces and the matrices of the extension space must be matched in their representation to make sure that the outcome will be subspaces with the required properties. To make these extensions and/or expansions simple to explain we will use certain representations of 2-subspaces and 3-subspaces of ${{\mathbb{F}}}_q^4$, and 2-subspaces and 3-subspaces of ${{\mathbb{F}}}_q^6$. These representations will also help to verify the correctness of the construction. For these representations we form an order between the vectors of length 4 of ${{\mathbb{F}}}_q^4$. For simplicity we will use the standard lexicographic order from the smallest to the largest 