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?
