Given two vector spaces $V_1$ and $V_2$, $V_1 \otimes V_2$ is called the tensor product of $V_1$ and $V_2$ and it is defined as follows:
Let $F(V_1 \times V_2)$ be the free vector space generated by $V_1 \times V_2$, and let $K \subset F(V_1 \times V_2)$ be the subspace of $F(V_1 \times V_2)$ generated by the elements \(\begin{align*} &(v_1 + v_1', v_2) - (v_1,v_2) - (v_1', v_2)\\ &(v_1, v_2+v_2') - (v_1,v_2)- (v_1, v_2') \\ &(\lambda v_1, v_2) - \lambda (v_1, v_2) \\ &(v_1, \lambda v_2) - \lambda(v_1,v_2)\end{align*}\) for all $v_1,v_1’\in V_1$, $v_2, v_2’ \in V_2$, and $\lambda \in \mathbb R$. The tensor product $V_1 \otimes V_2$ is the quotient vector space \(V_1 \otimes V_2 = F(V_1 \times V_2)/K.\)
There is also a definition in terms of a universal property.