Question

Suppose T in L(V) and U is a subspace of V.
Prove that if U subset of null T, then U is invariant under T.

Answer 1

Recall : A subspace W of a vector space V is called invariant under T if T(W)$\subset$ W

Since U is given to be subspace of vector space V

So, 0 must belong to U

Now again U is subset of null(T)

So, T(0)=0

and T(u)=0 $\forall$ u$\in$ U.

Hence T(u)=0 $\in$ U.

Hence, T(U)$\subset$ U.

So, U is invariant under T.