To show that the **null space** \( N(T) \) of a linear transformation \( T \) is a subspace of the domain of \( T \), we need to verify that \( N(T) \) satisfies the three defining properties of a subspace: 

1. **Contains the zero vector**.
2. **Closed under addition**.
3. **Closed under scalar multiplication**.

The null space of a transformation \( T: V \to W \) is defined as:

\[
N(T) = \{ v \in V \mid T(v) = 0 \}.
\]

### 1. **Contains the zero vector**:
We need to check that the zero vector of the domain \( V \) (denoted as \( 0_V \)) is in \( N(T) \). Since \( T \) is linear, we know that:

\[
T(0_V) = 0_W,
\]

where \( 0_W \) is the zero vector in \( W \). Therefore, \( 0_V \in N(T) \), so the null space contains the zero vector.

### 2. **Closed under addition**:
We need to verify that for any \( u, v \in N(T) \), the vector \( u + v \) is also in \( N(T) \). If \( u \in N(T) \) and \( v \in N(T) \), then:

\[
T(u) = 0_W \quad \text{and} \quad T(v) = 0_W.
\]

Because \( T \) is linear, we have:

\[
T(u + v) = T(u) + T(v) = 0_W + 0_W = 0_W.
\]

Thus, \( u + v \in N(T) \), so \( N(T) \) is closed under addition.

### 3. **Closed under scalar multiplication**:
We need to show that for any scalar \( c \in \mathbb{F} \) (where \( \mathbb{F} \) is the underlying field of the vector space) and any vector \( v \in N(T) \), the scalar multiple \( c v \) is also in \( N(T) \). If \( v \in N(T) \), then:

\[
T(v) = 0_W.
\]

By the linearity of \( T \), we have:

\[
T(c v) = c T(v) = c \cdot 0_W = 0_W.
\]

Thus, \( c v \in N(T) \), so \( N(T) \) is closed under scalar multiplication.

### Conclusion:
Since \( N(T) \) contains the zero vector, is closed under addition, and is closed under scalar multiplication, it satisfies the three conditions to be a subspace of the domain of \( T \). Therefore, \( N(T) \) is a subspace of \( V \), the domain of \( T \).

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Would you like more details or have any further questions?

Here are 5 related questions to think about:

1. What is the geometric interpretation of the null space?
2. How is the dimension of the null space related to the rank of a transformation?
3. Can the null space be a trivial subspace? When does this occur?
4. How does the null space of a matrix relate to solutions of a homogeneous system?
5. How is the null space affected by changes in the linear transformation?

**Tip:** The **rank-nullity theorem** connects the dimension of the null space to the dimension of the image (rank) of the transformation.

Understand how to prove that the null space N(T) of a linear transformation T is a subspace of its domain. The step-by-step solution covers key concepts in linear algebra, including subspace criteria and the linearity of transformations. Ideal for undergraduate students studying linear algebra.

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