(405) What is a tensor product in simple words? – Quora

For any two vector spaces $V$ and $W$ , the tensor product $V ⨂ W$ is the space of bilinear functions on $V \times W$ .

If $V$ and $W$ are inner product spaces then for any $v \in V$ and $w \in W$ we can define the pure tensor $v \otimes w$ in $V ⨂ W$ by $v \otimes w (v *, w *) = (v \cdot v *) (w \cdot w *)$ also often written by physicists as $⟨ v | v * ⟩ ⟨ w | w * ⟩$ . But there are also elements of $V ⨂ W$ that are not of the pure tensor form.

For example if $v_{1} \otimes w_{1} + v_{2} \otimes w_{2}$ could be written in the form $(a_{1} v_{1} + a_{2} v_{2}) \otimes (b_{1} w_{1} + b_{2} w_{2})$ , then for all $v \in V$ and $w \in W$ , we’d need $\begin{aligned} (a_{1} v_{1} + a_{2} v_{2}) \otimes (b_{1} w_{1} + b_{2} w_{2}) (v, w) \\ = (a_{1} v_{1} \cdot v) (b_{1} w_{1} \cdot w) + (a_{1} v_{1} \cdot v) (b_{2} w_{2} \cdot w) \\ + (a_{2} v_{2} \cdot v) (b_{1} w_{1} \cdot w) + (a_{2} v_{2} \cdot v) (b_{2} w_{2} \cdot w) \\ = (v_{1} \cdot v) (w_{1} \cdot w) + (v_{2} \cdot v) (w_{2} \cdot w) \end{aligned}$ .

But this is only true if $a_{1} b_{1} = a_{2} b_{2} = 1$ and $a_{1} b_{2} = a_{2} b_{1} = 0$ , but if one of $a_{1}$ or $b_{2}$ is zero then one of $a_{1} b_{1}$ or $a_{2} b_{2}$ must be also.

Source: (405) What is a tensor product in simple words? – Quora

Leave a Reply Cancel reply