Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Say that two vectors from $U$, $\vect{x}$ and $\vect{y}$, are related exactly when $\lt{T}{\vect{x}}=\lt{T}{\vect{y}}$ in $V$. Prove the three properties of an equivalence relation on $U$: (a) for any $\vect{x}\in U$, $\vect{x}$ is related to $\vect{x}$, (b) if $\vect{x}$ is related to $\vect{y}$, then $\vect{y}$ is related to $\vect{x}$, and (c) if $\vect{x}$ is related to $\vect{y}$ and $\vect{y}$ is related to $\vect{z}$, then $\vect{x}$ is related to $\vect{z}$.