Answer to Question #121535 in Differential Geometry | Topology for Brain

Question #121535
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;
1
Expert's answer
2020-06-14T16:33:15-0400

Operator "T\\in End(F^2)" i.e "T:F^2\\longrightarrow F^2" such that "T(x,y)=(y,0)"

Let,

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

Note that, for any "u\\in U" ,thus "u=(x_1,0)" for some "x_1\\in F"

"T(u)=(0,0)\\hspace{1cm}(\\because" By definition of "T)" which implies "T(u)\\in U" ,Hence "T" is invariant under "U" .

Also note that "u" was arbitrary and "u\\in U\\implies u\\in F^2" but "T|_U:U\\longrightarrow U" ,thus

"T|_U(u)=T(u)=0\\implies T|_U=0"


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