Linear maps preserving ₚ-norm parallel vectors

Two vectors $x, y$ in a normed vector space are parallel if there is a scalar μ with |μ| = 1 such that \|x+μ y\| = \|x\| + \|y\|; they form a triangle equality attaining (TEA) pair if $\|x+y\| = \|x\| + \|y\|$. In this paper, we characterize linear maps on Fⁿ=Rⁿ or Cⁿ, equipped with the ₚ-norm for p ∈ [1, ∞], preserving parallel pairs or preserving TEA pairs. Indeed, any linear map will preserve parallel pairs and TEA pairs when 1< p <∞. For the ₁-norm, TEA preservers form a semigroup of matrices in which each row has at most one nonzero entries; adding rank one matrices to this semigroup will be the semigroup of parallel preserves. For the _∞-norm, a nonzero TEA preserver, or a parallel preserver of rank greater than one, is always a multiple of an _∞-norm isometry, except when Fⁿ = R². We also have a characterization for the exceptional case. The results are extended to linear maps of the infinite dimensional spaces ₁(Λ), c₀(Λ) and _∞(Λ).