Answer to Question #121534 in Abstract Algebra for Brain

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 iff T(W)$\subset$ W

Since U is given to be subspace of V so, 0 belongs to U

Now again U is subset of null space of V

Therefore , T(0)=0

For all u$\in$ U

We have T(u)=0 (because U is subset of null(T)

$\forall$ u$\in$ U , T(u)=0$\in$ U

So, T(U)$\subset$ U

Hence , U is invariant under T