Show that $U:=\left\{x_{1} \otimes v | v \in V\right\}$ is a subspace

Question

Let $V$ be vector space with basis $\{x_1,...,x_m\}$. I want to show that $$U:=\left\{x_{1} \otimes v | v \in V\right\}$$ Is a subspace of $V\otimes V$.
My attempt: To show that $0\in U$, I pick $v=0$: $$x_{1} \otimes 0=0, \quad \Rightarrow 0 \in U$$ And since $V$ is a vector space: $$v_{1}, v_{2} \in V \Rightarrow \alpha v_{1}+v_{2} \in V$$ Where $\alpha$ is a scalar. I can then write: $$x_{1} \otimes \alpha_{1} v_{1}+x_{1} \otimes v_{2}=x_{1} \otimes\left(\alpha_{1} v_{1}+v_{2}\right) \in U$$ Which shows that $U$ is a subspace of $V\otimes V$. Any flaws in my proof?

Answer 1

Your proof is mostly correct. However, keep in mind that to show that $U$ is a subspace, we must show that $\alpha_1(x_1 \otimes v_1) + (x_1 \otimes v_2)$, so your equation at the end should have an "extra step".