- #1
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1) let S:U->V T:V->W be linear operators, show that: (ToS)^t=S^toT^t.
2) let T:V->U be linear and u belongs to U, show that u belongs to Im(T) or that there exist [tex]\phi\inV*[/tex] such that [tex]T^{t}(\phi)=0[/tex] and [tex]\phi(u)=1[/tex]
about the first question here what i tried to do:
[tex](ToS)^{t}(\phi(v))=\phi o(ToS)(v)=\phi(T(S(v))=T^{t}(\phi(S(v))=T^{t}(S^{t}(\phi(v))=T^{t}oS^{t}[/tex]
about the second question the first part (of u belongs to ImT) i did it (simply the defintion of ImT), but i don't know how to to approach the latter.
ofcourse i have [tex]T^{t}(\phi(v))=\phi(T(v))=\phi(u)[/tex] where v belongs to V, but other than this i don't know how to proceed.
2) let T:V->U be linear and u belongs to U, show that u belongs to Im(T) or that there exist [tex]\phi\inV*[/tex] such that [tex]T^{t}(\phi)=0[/tex] and [tex]\phi(u)=1[/tex]
about the first question here what i tried to do:
[tex](ToS)^{t}(\phi(v))=\phi o(ToS)(v)=\phi(T(S(v))=T^{t}(\phi(S(v))=T^{t}(S^{t}(\phi(v))=T^{t}oS^{t}[/tex]
about the second question the first part (of u belongs to ImT) i did it (simply the defintion of ImT), but i don't know how to to approach the latter.
ofcourse i have [tex]T^{t}(\phi(v))=\phi(T(v))=\phi(u)[/tex] where v belongs to V, but other than this i don't know how to proceed.