= Solution
The finite-dimensional <inner product space> has the <orthogonal decomposition> $U=W\oplus W^\perp$. Define $\tau(w+z)=w$ for $w\in W$, $z\in W^\perp$. This is a <linear map> with image $W$, satisfies $\tau^2=\tau$, and is self-adjoint by taking <inner products> between decomposed vectors. Hence it is an <orthogonal projection>.
Any other <orthogonal projection> with image $W$ has kernel $W^\perp$ by part (a); it is identity on $W$ by idempotence and zero on its kernel. Its values are therefore forced to agree with $\tau$, proving uniqueness.
If $\tau_2\tau_1=0$, every $w_1\in W_1$ has $w_1=\tau_1w_1$, so $\tau_2w_1=0$ and $w_1\in W_2^\perp$. Conversely, if $W_1\subset W_2^\perp$, every image $\tau_1u$ is killed by $\tau_2$. Therefore
$$
\boxed{\tau_2\circ\tau_1=0\quad\Longleftrightarrow\quad W_1\perp W_2.}
$$
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