Tensor product

$V⨂W$ is the tensor product of Vector Spaces $V$ and $W$.

It is built by constructing the Cartesian Product $Z:=\{v⨂w:v\in V,w\in W\}$, where $v⨂w$ can be seen as representing the pair $(v,w)$ and taking the Quotient Space of $Z$ with the relations:

1. $v_1⨂w+v_2⨂w=(v_1+v_2)⨂w$
2. $v⨂w_1+v⨂w_2=v⨂(w_1+w_2)$
3. $\lambda (v⨂w)=(\lambda v)⨂w=v⨂(\lambda w)$ In the above $\lambda \in \mathbb{F}$, where $V$ and $W$ are modules over the Field $\mathbb{F}$.

Examples

• $V⨂\mathbb{F}\cong V$ for any module $V$ over a Field $\mathbb{F}$.
• ${\mathbb{R}}^m⨂{\mathbb{R}}^n\cong {\mathbb{R}}^{mn}$.

Properties

Universal property of tensor products

See more:

• The tensor product demystified