The  canonical  line  bundle over $\mathbb{C}P^{n}$ is  denoted by $\ell_{n}$. It is  well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic  bundles  as two  topological vector  bundles over $\mathbb{C}P^{1}$.The  argument is  explained in "K-theory  and  vector  bundles"  by  Allen  Hatcher. In fact the  argument is  based on the  fact that the  following two  maps from $S^{1}$ to $GL_{2}(\mathbb{C})$ are  homotopic  maps
$$\begin{pmatrix}z & 0\\ 0&z \end{pmatrix}\;\;\text{and}\begin{pmatrix}z^{2} & 0\\ 0&1 \end{pmatrix}$$

**(A topological question):**

> How can we  compare $(\ell_{n}\otimes \ell_{n})\oplus1 $  with $\ell_{n}\oplus\ell_{n}$. We know  that their  restriction to $\mathbb{C}P^{1}\subset \mathbb{C}P^{n}$ are isomorphic. So what is  the  **difference element**  as  an  element of the relative  K-theory $K(\mathbb{C}P^{n},\mathbb{C}P^{1})$?


**(An  algebra  geometric  question):**

> Is the  isometry $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ valid  as an isomorphism of vector  bundles in   the  context  of  algebraic geometry?