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 $$|\phi\rangle \langle \phi|$$?