Tensor Products: • The tensor product of two vector spaces, V ⊗ W , is a vector space constructed such that: – If dim(V ) = n and dim(W ) = m, then dim(V ⊗ W ) = n · m. – Elements of V ⊗ W are expressed as linear combinations of v ⊗ w, where v ∈ V and w ∈ W . • The operation ⊗ satisfies bilinearity: (av +bu)⊗w = a(v ⊗w)+b(u⊗w), v ⊗(aw+bz) = a(v ⊗w)+b(v ⊗z). Properties of Tensor Products • Tensor products distribute over direct sums: (V1 ⊕ V2 ) ⊗ W ∼ = (V1 ⊗ W ) ⊕ (V2 ⊗ W ). • Associativity: U ⊗ (V ⊗ W ) ∼ = (U ⊗ V ) ⊗ W. • Commutativity (up to isomorphism): V ⊗W ∼ = W ⊗ V. Examples and Applications Example 1: Tensor Products of Basis Elements • Let V = R2 and W = R3 , with bases {e1 , e2 } and {f1 , f2 , f3 }, respectively. • The basis for V ⊗ W consists of all elements ei ⊗ fj : {e1 ⊗ f1 , e1 ⊗ f2 , e1 ⊗ f3 , e2 ⊗ f1 , e2 ⊗ f2 , e2 ⊗ f3 }. • Therefore, dim(V ⊗ W ) = 2 · 3 = 6. Example 2: Tensor Product of Matrices • Compute the tensor product of the matrices: 1 2 0 5 A= , B= . 3 4 6 7 • The result is: Visualizing Tensor Products Diagram 1: Basis Elements in Tensor Products e1 ⊗ f1 e1 e1 ⊗ f2 e 2 ⊗ f1 e 1 ⊗ f3 e2 f1 f2 e2 ⊗ f2 e2 ⊗ f3 f3 Diagram 2: Tensor Product of Matrices 1·B 3·B Row 1 Row 2 2 2·B 4·B