If we are given the local trivialities of a holomorphic vector bundle then by definition we can write down the transition matrix of that vector bundle.

In some very natural situations, we are not given local trivialities of that vector bundle but are given some description for it. The question is how we can write down the transition matrices from what are given. I will illustrate the question by two examples.

Example 1: This is a classical example. Let $J$ be the universal line bundle over $\mathbb{CP}^n$, which is given by $J=\{(z,\theta )\in \mathbb{CP}^n\times \mathbb{C}^{n+1}:~\theta \mbox{ is in the line generated by }z\}$. In this case, we can choose a meromorphic section $s$ for $J$ by $([z_0:z_1:\ldots :z_{n}],(1,z_1/z_0,\ldots ,z_n/z_0))$. The divisor of $s$ is $-z_0$, and hence we have $J=[-H]$.

Example and Question 2: We consider a similar construction. Let $W=\mathbb{CP}^n\times \mathbb{CP}^n-\Delta $ where $\Delta \subset \mathbb{CP}^n\times \mathbb{CP}^n$ is the diagonal. Let $V=(z_1,z_2,\theta )\in W\times \mathbb{C}^{n+1}$ so that $\theta$ is in the plane generated by $z_1$ and $z_2$. Then $V$ is a holomorphic vector bundle of rank $2$ on $W$. Then what is the transition matrix for this vector bundle?

I tried to think about this question off and on but still get stuck. Any help or hint is very helpful.

If we are given the local trivialities of a holomorphic vector bundle then by definition we can write down the transition matrix of that vector bundle.

In some very natural situations, we are not given local trivialities of that vector bundle but are given some description for it. The question is how we can write down the transition matrices from what are given. I will illustrate the question by two examples.

Example 1: This is a classical example. Let $J$ be the universal line bundle over $\mathbb{CP}^n$, which is given by $J=\{(z,\theta )\in \mathbb{CP}^n\times \mathbb{C}^{n+1}:~\theta \mbox{ is in the line generated by }z\}$. In this case, we can choose a meromorphic section $s$ for $J$ by $([z_0:z_1:\ldots :z_{n}],(1,z_1/z_0,\ldots ,z_n/z_0))$. The divisor of $s$ is $-z_0$, and hence we have $J=[-H]$.

Example and Question 2: We consider a similar construction. Let $W=\mathbb{CP}^n\times \mathbb{CP}^n-\Delta $ where $\Delta \subset \mathbb{CP}^n\times \mathbb{CP}^n$ is the diagonal. Let $V=(z_1,z_2,\theta )\in W\times \mathbb{C}^{n+1}$ so that $\theta$ is in the plane generated by $z_1$ and $z_2$. Then $V$ is a holomorphic vector bundle of rank $2$ on $W$. Then what is the transition matrix for this vector bundle?

I tried to think about this question off and on but still get stuck. Any help or hint is very helpful.

Addition: My comment after David's comment was not totally correct, so I add this into my question. So in my Questions (either Example 1 or 2) above, you are free to choose any open cover for the base space, and my question is can you write down a trivial for the vector bundle?

If we are given the local trivialities of a holomorphic vector bundle then by definition we can write down the transition matrix of that vector bundle.

In some very natural situations, we are not given local trivialities of that vector bundle but are given some description for it. The question is how we can write down the transition matrices from what are given. I will illustrate the question by two examples.

Example 1: This is a classical example. Let $J$ be the universal line bundle over $\mathbb{CP}^n$, which is given by $J=\{(z,\theta )\in \mathbb{CP}^n\times \mathbb{C}^{n+1}:~\theta \mbox{ is in the line generated by~}z\}$. In this case, we can choose a meromorphic section $s$ for $J$ by $([z_0:z_1:\ldots :z_{n}],(1,z_1/z_0,\ldots ,z_n/z_0))$. The divisor of $s$ is $-z_0$, and hence we have $J=[-H]$.

Example and Question 2: We consider a similar construction. Let $W=\mathbb{CP}^n\times \mathbb{CP}^n-\Delta $ where $\Delta \subset \mathbb{CP}^n\times \mathbb{CP}^n$ is the diagonal. Let $V=(z_1,z_2,\theta )\in W\times \mathbb{C}^{n+1}$ so that $\theta$ is in the plane generated by $z_1$ and $z_2$. Then $V$ is a holomorphic vector bundle of rank $2$ on $W$. Then what is the transition matrix for this vector bundle?

I tried to think about this question off and on but still get stuck. Any help or hint is very helpful.

If we are given the local trivialities of a holomorphic vector bundle then by definition we can write down the transition matrix of that vector bundle.

In some very natural situations, we are not given local trivialities of that vector bundle but are given some description for it. The question is how we can write down the transition matrices from what are given. I will illustrate the question by two examples.

Example 1: This is a classical example. Let $J$ be the universal line bundle over $\mathbb{CP}^n$, which is given by $J=\{(z,\theta )\in \mathbb{CP}^n\times \mathbb{C}^{n+1}:~\theta \mbox{ is in the line generated by }z\}$. In this case, we can choose a meromorphic section $s$ for $J$ by $([z_0:z_1:\ldots :z_{n}],(1,z_1/z_0,\ldots ,z_n/z_0))$. The divisor of $s$ is $-z_0$, and hence we have $J=[-H]$.

Example and Question 2: We consider a similar construction. Let $W=\mathbb{CP}^n\times \mathbb{CP}^n-\Delta $ where $\Delta \subset \mathbb{CP}^n\times \mathbb{CP}^n$ is the diagonal. Let $V=(z_1,z_2,\theta )\in W\times \mathbb{C}^{n+1}$ so that $\theta$ is in the plane generated by $z_1$ and $z_2$. Then $V$ is a holomorphic vector bundle of rank $2$ on $W$. Then what is the transition matrix for this vector bundle?

I tried to think about this question off and on but still get stuck. Any help or hint is very helpful.