Show that T - I is a projection. Came across a question where:


I could solve (a) and (b), but have no clue what (c) means.
Please advice.

By definition:

A projection is a linear transformation $P$ from a vector space to itself such that $P^2 = P$.

You have found that 
$$
T=\begin{bmatrix}
1&0&0\\
1&1&1\\
1&0&2
\end{bmatrix}
$$
 now you have to prove that:
$$
P=(T-I)=
\begin{bmatrix}
1&0&0\\
1&1&1\\
1&0&2
\end{bmatrix}-
\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1
\end{bmatrix}
=\begin{bmatrix}
0&0&0\\
1&0&1\\
1&0&1
\end{bmatrix}
$$
is such that $P^2=P$ (that is easy)
