To analyze the linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $T(a, b, c) = (a-b, a-b, 0)$, we need to determine its null space $N(T)$ and its range space $R(T)$.

Null Space $N(T)$

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

Given $T(a, b, c) = (a-b, a-b, 0)$, we need to solve: $(a - b, a - b, 0) = (0, 0, 0)$ This implies: $a - b = 0 \quad \text{and} \quad 0 = 0$ From $a - b = 0$, we get $a = b$.

Thus, the null space $N(T)$ consists of all vectors of the form $(a, a, c)$, where $a$ and $c$ are arbitrary real numbers. Therefore: $N(T) = \{ (a, a, c) \mid a, c \in \mathbb{R} \}$ This can be expressed as the span of the vectors $(1, 1, 0)$ and $(0, 0, 1)$: $N(T) = \text{span} \{ (1, 1, 0), (0, 0, 1) \}$

Range Space $R(T)$

The range space $R(T)$ is the set of all vectors in $\mathbb{R}^3$ that can be written as $T(a, b, c)$ for some $(a, b, c) \in \mathbb{R}^3$.

Given $T(a, b, c) = (a - b, a - b, 0)$, we observe that the first two components are always equal, and the third component is always zero. Therefore, any vector in the range of $T$ has the form $(x, x, 0)$.

Thus, the range space $R(T)$ consists of all vectors of the form $(x, x, 0)$, where $x$ is an arbitrary real number: $R(T) = \{ (x, x, 0) \mid x \in \mathbb{R} \}$ This can be expressed as the span of the vector $(1, 1, 0)$: $R(T) = \text{span} \{ (1, 1, 0) \}$

Summary

1. The null space $N(T)$ is $\text{span} \{ (1, 1, 0), (0, 0, 1) \}$, which is a 2-dimensional subspace of $\mathbb{R}^3$.
2. The range space $R(T)$ is $\text{span} \{ (1, 1, 0) \}$, which is a 1-dimensional subspace of $\mathbb{R}^3$.

Options Analysis

Given the analysis above, the correct option that describes $N(T)$ and $R(T)$ is:

$\boxed{\text{N(T) is 2-dimensional and R(T) is 1-dimensional}}$

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Do you want further details or have any questions about this problem? Here are some related questions for you:

1. What is the dimension of the kernel of a linear transformation?
2. How do you find the null space of a linear transformation?
3. What is the rank-nullity theorem?
4. Can you provide an example of another linear transformation and its null and range spaces?
5. How do you determine if a linear transformation is onto?
6. What is the significance of the dimension of the null space?
7. How can you represent a linear transformation using a matrix?
8. What is the relationship between the null space and the invertibility of a linear transformation?

Tip: Always check if the null space includes all vectors that map to the zero vector by the given transformation.