Question

Define an operator T in End(F^2) by T(x,y)= (y,0) Let
U = {(x,0) | x in F}. Show that
U is invariant under T and T |U is the 0 operator on U;

Answer 1

Given that, $T$ is an operator such that $T\in End(F^2)$ and defined by

$$T(x,y)= (y,0)$$

And,

$$U := \{(x,0) | x \in F\}$$

Since, if $U$ is invariant under $T$ ,thus

$$T(U)\subset U$$

In this case, suppose $u\in U\implies u=(x,0)$ for some $x\in F$ ,Thus by definition

$$T(u)=T(x,0)=(0,0)\in U \hspace{1cm}(\because 0\in U)\\ \implies T(U)\subset U$$

Now, we have to show the restriction of $T$ under $U$ i.e

$$T|_U:U\longrightarrow U$$

is operator.

Clearly, for all

$$v\in U\implies v\in F^2 \implies T|_U(v)=T(v)=0\\ \implies T|_U=0$$

Hence we are done.