Section Example.

(** - Example 26 on page 20 is actually WRONG, or at least incomplete: if the field is $\mathbf Z/2\mathbf Z = \{0,1\}$, where $-1=1$, then symmetric and antisymmetric matrices coincide, and $M_{nn}$ cannot be split in just the symmetric and antisymmetric part. Hence the extra condition that $1+1\neq0$: *)
Lemma matrixspace_sym_antisym :
forall (n : nat) (F : field),
~ one +' one =' (zero F) in _ ->
form_direct_sum (symm_subspace n F) (antisym_subspace n F).