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.