# Thread: Restriction on a linear operator

1. ## Restriction on a linear operator

Prove that the restriction of a linear operator T to a T-invariant subspace is a linear operator on that subspace.

For this proof, I'm not sure if I fully understand the problem.

Now, let W be a vector space, then T(W) is a subset of W. So this is the restriction. So I have to show that as a linear operator? How?

2. Hello

The subspace considered is not necessarily the whole space $\displaystyle W$, it is a subspace which is $\displaystyle T$-invariant. If we call it $\displaystyle V\subset W$, it means that for all $\displaystyle x\in V,\,T(x) \in V$. Now you've to show that the restriction of $\displaystyle T$ to $\displaystyle V$ (let's call it $\displaystyle T_v$) is a linear operator, that is to say :
• $\displaystyle \forall x,\,y\in V,\,T_v(x)+T_v(y)=T_v(x+y)$
• $\displaystyle \forall \lambda\in\mathbb{K},\,x\in V,\,T_v(\lambda x)=\lambda T_v(x)$

Aknoledegded that $\displaystyle T$ is a linear operator on $\displaystyle W$, it should not be a problem.