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Confusion regarding projection operator

Suppose we have a qutrit with the state vector $|\psi\rangle = a_0|0\rangle + a_1|1\rangle + a_2|2\rangle$, and we want to project its state onto the subspace having the basis $\{|0\rangle,|2\rangle\}$, I know the projection operator would be written like: $1|0\rangle \langle0| + 0|1\rangle\langle 1| + 1|2\rangle\langle2|$.

I'm having a few confusions here. Does $|0\rangle \langle 0|$ represent a tensor product between $[1 \ 0 \ 0]^{T}$ and $[1 \ 0 \ 0]$ ? Or is it just matrix multiplication? Also, I thought that we must always be able to write a projection operator in the form $|\phi\rangle \langle \phi|$ where $|\phi\rangle$ is a possible state of a qutrit. But how to represent $1|0\rangle \langle0| + 0|1\rangle\langle 1| + 1|2\rangle\langle2|$ in the form $|\phi\rangle \langle \phi|$?