Let   T: R^3 --->  R^3  be defined by T (a,b,c)= (a-b, a-b,0).  Here N(T) and R(T) denote the null space and Range space of T respectively. Choose the correct option.

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) = \

Explore the linear transformation T(a, b, c) = (a - b, a - b, 0) from R^3 to R^3. Learn about its null space and range space, including detailed solutions and interpretations. Suitable for advanced undergraduate students and beyond.

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