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## Homework Statement

T : R[itex]^{3}[/itex] -> R[itex]^{3}[/itex] is a linear transformation. We need to prove the equivalence of the three below statements.

i) R[itex]^{3}[/itex] = ker(T) [itex]\oplus[/itex] im(T);

ii) ker(T) = ker(T[itex]^{2}[/itex]);

iii) im(T) = im(T[itex]^{2}[/itex]).

## Homework Equations

R[itex]^{3}[/itex] = ker(T) [itex]\oplus[/itex] im(T), if for all v [itex]\in[/itex] R[itex]^{3}[/itex] there exists x [itex]\in[/itex] ker(T) and y [itex]\in[/itex] im(T) such that v = x + y, and ker(T) [itex]\bigcap[/itex] im(T) = {0}

ker(T) = {

__x__[itex]\in[/itex]R[itex]^{3}[/itex] : T(

__x__)=0}

im(T) = {

__w__[itex]\in[/itex]R[itex]^{3}[/itex] :

__w__=f(

__x__), x[itex]\in[/itex]R[itex]^{3}[/itex]}

## The Attempt at a Solution

I really have no idea how to show these statements are equivalent. Can someone also clarify the linear mapping T[itex]^{2}[/itex]?

Thanks.