We identify the vector space tensor product $\mathbb{R}^{m} \otimes \mathbb{R}^{n}$ with $\mathbb{R}^{mn}$

Let $X$ be the space of all non zero simple tensors $X=\{a\otimes b \mid a\in \mathbb{R}^{n} \setminus \{0\}, \;b\in \mathbb{R}^{m} \setminus \{0\}\}$. Let $\pi:\mathbb{R}^{mn}\setminus \{0\} \to \mathbb{R}P^{(mn-1)} $ be the natural projection. Put $PX=\pi (X)$$

Is the tautological line bundle restricted to $PX$, a trivial bundle? Is the following bundle $(E,X,q))$ a trivial bundle over $X$:

$E=\{(x\otimes y, T) \mid T:E_{x} \to E_{y} \;\; \text{is a linear map }$ where $E_{x}= \{v\in \mathbb{R}^{n} \mid v.x=0\} $

$X$ is the space simple tensors and $q$ is the obvious projection.

We identify the vector space tensor product $\mathbb{R}^{m} \otimes \mathbb{R}^{n}$ with $\mathbb{R}^{mn}$

Let $X$ be the space of all non zero simple tensors $X=\{a\otimes b \mid a\in \mathbb{R}^{n} \setminus \{0\}, \;b\in \mathbb{R}^{m} \setminus \{0\}\}$. Let $\pi:\mathbb{R}^{mn}\setminus \{0\} \to \mathbb{R}P^{(mn-1)} $ be the natural projection. Put $PX=\pi (X)$

Is the tautological line bundle restricted to $PX$, a trivial bundle? Is the following bundle $(E,X,q))$ a trivial bundle over $X$:

$E=\{(x\otimes y, T) \mid T:E_{x} \to E_{y} \;\; \text{is a linear map }$ where $E_{x}= \{v\in \mathbb{R}^{n} \mid v.x=0\} $

$X$ is the space simple tensors and $q$ is the obvious projection.