Originally Posted by

**Last_Singularity** For these problems:

1. Define the annihilator of $\displaystyle S^0$ as

$\displaystyle S^0 = \{f \in V*: f(x) = 0 \forall x \in S\}$, where $\displaystyle S$ is a subset of finite-dimensional vector space $\displaystyle V$.

2. Let $\displaystyle V,W$ be finite-dimensional vector spaces with ordered bases $\displaystyle \beta, \gamma$ respectively. Then for any linear mapping $\displaystyle T: V \rightarrow W$, the mapping $\displaystyle T^t: W* \rightarrow V*$ is defined by $\displaystyle T^t(g)=gT$ for all $\displaystyle g \in W*$ with the property that $\displaystyle [T^t]_{\gamma *}^{\beta *} = ([T]_\beta^gamma)^t$.

Question 1: Suppose that $\displaystyle W$ is a finite-dimensional vector space and $\displaystyle T: V \rightarrow W$ is linear. Prove that $\displaystyle N(T^t) = (R(T))^0$.

Question 2: Let $\displaystyle T$ be a linear operator on $\displaystyle V$ and let $\displaystyle W$ be a subspace of $\displaystyle V$. Prove that $\displaystyle W$ is $\displaystyle T$-invariant if and only if $\displaystyle W^0$ is $\displaystyle T^t$-invariant.