let $T_1 : V \to W $ and $T_2 : V \to W $
how to show that $Kernel(T_1) \cap Kernel(T_2) \subset kernel(T1+T2) $?
Kernel of intersection of T1 and T2 is contained in kernel of (T1 +T2)
0
$\begingroup$
linear-algebra
vector-spaces
-
1Just take an element in both, and check it gives $0$ when applying $T_1+T_2$. – 2017-02-11
1 Answers
2
Let $ x \in \ker T_1 \cap \ker T_2 $ be given. Then $T_1(x) = T_2(x) = 0$, so $$(T_1 + T_2)(x) = T_1(x) + T_2(x) = 0+0=0, $$ and thus $x \in \ker(T_1 + T_2) $