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 TEnd(F2)T\in End(F^2) i.e T:F2F2T:F^2\longrightarrow F^2 such that T(x,y)=(y,0)T(x,y)=(y,0)

Let,

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

Note that, for any uUu\in U ,thus u=(x1,0)u=(x_1,0) for some x1Fx_1\in F

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

Also note that uu was arbitrary and uU    uF2u\in U\implies u\in F^2 but TU:UUT|_U:U\longrightarrow U ,thus

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


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