Answer to Question #121523 in Linear Algebra for Tau

Question

Suppose, T in Hom(V,W) then prove that Nullspace of T is subspace of V

Answer 1

Let $v_1$  and $v_2$  belong to the nullspace of $T$ . Then $T(v_1) = 0$  and $T(v_2) = 0$  by the definition of nullspace. Then $T(v_1+v_2) = T(v_1) + T(v_2) = 0+0 = 0$  because $T$  is linear, and $v_1+v_2$  belongs to the nullspace of $T$ . Therefore, the nullspace of $T$  is closed under addition.

Since $T(0) = 0$  because $T$  is linear, the nullspace of $T$  contains .

Let $k$  be a scalar and $v$  belong to the nullspace of $T$ . Then $T(v) = 0$  by the definition of nullspace. Then $T(kv) = k\cdot T(v) = k\cdot 0 = 0$  because $T$  is linear, and $kv$  belongs to the nullspace of $T$ . Therefore, the nullspace of $T$  is closed under multiplication by a scalar.

By the definition of subspace, the nullspace of $T$  is a subspace of $V$ .