Let's analyze the linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $T(a, b, c) = (a - b, a - b, 0)$.

Null Space $N(T)$

The null space $N(T)$ consists of all vectors $(a, b, c) \in \mathbb{R}^3$ such that $T(a, b, c) = (0, 0, 0)$.

$T(a, b, c) = (a - b, a - b, 0) = (0, 0, 0)$

This gives us the system of equations: $a - b = 0$ $a - b = 0$ $0 = 0$

From the first equation, we have $a = b$. The third equation is always satisfied, and $c$ can be any real number. So, the null space is:

$N(T) = \{ (a, a, c) \mid a, c \in \mathbb{R} \}$

This can be written as: [ N(T) = \