Tensor Product

Jun 4, 2017

1 - Introduction to the tensor product

2 - Tensor product of linear maps

$Tensor product of vector spaces Definition Main article: Tensor product of two modules on a commutative ring. Theorem and definition. Let E and F be two vector spaces on a commutative field K. There exists a vector space, denoted E ⊗ F { displaystyle E otimes F} { displaystyle E otimes F}, and a bilinear map Φ: E × F → E ⊗ F { displaystyle phi: E times F rightarrow E Y) = x ⊗ y { displaystyle phi (x, y) = x otimes y} (G), and for every bilinear map g of E × F in G, there exists one and only one linear map g ~ { displaystyle { tilde {g}}} { Tilde g} of E ⊗ F { displaystyle E otimes F} { displaystyle E otimes F} in G such that G = g ~ ∘ φ { displaystyle g = { tilde {g}} circ phi} g = { tilde g} circ phi or else ∀ x ∈ E, ) = G ~ (x ⊗ y). { Displaystyle forall x in E, y in F, g (x, y) = { tilde {g} F, g (x, y) = { tilde {g}} (x otimes y). Moreover, this space is unique (with isomorphism close). The space E ⊗ F { displaystyle E otimes F} { displaystyle E otimes F} is the tensor product of E and F, and x ⊗ y { displaystyle x otimes y} x otimes is the product Tensor of x and y. If (i) i ∈ I { displaystyle (e_ {i}) {{i in I}} and (fj) (I, j)) are respectively bases of E and F, then (ei ⊗ fj) (i, j) ∈ I (I, j)} {{i (i, j)} {{i} J) in I times J}} is a basis of E ⊗ F { displaystyle E otimes F} { displaystyle E otimes F}. In particular, if E and F are of finite dimension, (F) = mathrm {dim} (E) dimmers {m} (dim) { Displaystyle mathrm {dim} (E otimes F) = mathrm {dim} Technically, the existence and uniqueness theorem is a safeguard that makes it possible to be satisfied from the point of view of the bases.$