Null Space is a Subspace

The null space is a subspace.

Suppose $T \in L (V, W)$ . Then the null space of $T$ , $null T$ , is a subspace of $V$ .

Proof. Because $T$ is a linear map, $T (0) = 0$ (Linear Maps Take 0 to 0). Thus $0 \in null T$ .

Suppose $u, v \in null T$ . Then

T (u + v) = T u + T v = 0 + 0 = 0

Hence $u + v \in null T$ . Thus, $null T$ is closed under addition.

Suppose $u \in null T$ and $λ \in F$ . Then

T (λ u) = λ T u = λ 0 = 0

Hence, $λ u \in null T$ . Thus, $null T$ is closed under scalar multiplication.

We have shown that $null T$ is closed under scalar multiplication.

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