English
If pb is a power basis for L over K, then the finite sum of σ(pb.gen) over embeddings σ from L to E equals finrank_L F times the universal sum of pb.gen under AlgHom pb.
Русский
Если pb — степенной базис L над K, сумма σ(pb.gen) по вложениям σ равна finrank_L F умножить на обобщённую сумму pb.gen.
LaTeX
$$$\\sum_{\\sigma : F \\to E} σ(\\alpha) = \\operatorname{finrank}_L F \\cdot \\sum_{σ: L\\to_K E} σ(\\mathrm{gen})$$$
Lean4
theorem traceMatrix_of_matrix_mulVec [Fintype κ] (b : κ → B) (P : Matrix κ κ A) :
traceMatrix A (P.map (algebraMap A B) *ᵥ b) = P * traceMatrix A b * Pᵀ :=
by
refine AddEquiv.injective (transposeAddEquiv κ κ A) ?_
rw [transposeAddEquiv_apply, transposeAddEquiv_apply, ← vecMul_transpose, ← transpose_map,
traceMatrix_of_matrix_vecMul, transpose_transpose]
