tensor product of vector spaces (changes) in nLab

Context

Linear algebra

• Idea
• Definition
• References
• Related concepts

Definition

Given two vector spaces over some field kk, V 1,V 2∈Vect kV_1, V_2 \in Vect_k, their tensor product of vector spaces is the vector space denoted

V 1⊗ kV 2∈Vect V_1 \otimes_k V_2 \in Vect

whose elements are equivalence classes of formal linear combinations of tuples (v 1,v 2)(v_1,v_2) with v i∈V iv_i \in V_i, for the equivalence relation given by

(kv 1,v 2)∼k(v 1,v 2)∼(v 1,kv 2) (k v_1 , v_2) \;\sim\; k( v_1 , v_2) \;\sim\; (v_1, k v_2)

(v 1+v′ 1,v 2)∼(v 1,v 2)+(v′ 1,v 2) (v_1 + v'_1 , v_2) \; \sim \; (v_1,v_2) + (v'_1, v_2)

(v 1,v 2+v′ 2)∼(v 1,v 2)+(v 1,v′ 2) (v_1 , v_2 + v'_2) \; \sim \; (v_1,v_2) + (v_1, v'_2)

More abstractly this means that the tensor product of vector spaces is the vector space characterized by the fact that

1. it receives a bilinear map

   V 1×V 2⟶V 1⊗V 2 V_1 \times V_2 \longrightarrow V_1 \otimes V_2

   (out of the Cartesian product of the underlying sets)

2. any other bilinear map of the form

   V 1×V 2⟶V 3 V_1 \times V_2 \longrightarrow V_3

   factors through the above bilinear map via a unique linear map

   V 1×V 2 ⟶bilinear V 3 ↓ ↗ ∃!linear V 1⊗ kV 2 \array{ V_1 \times V_2 &\overset{bilinear}{\longrightarrow}& V_3 \\ \downarrow & \nearrow_{\mathrlap{\exists ! \, linear}} \\ V_1 \otimes_k V_2 }