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Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) Prove that

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The solution is give beiou imear qnstoa Hre be let T V Vw and W V SPaces mation Vee t0h betwee MAN (UHere let kez (T) = {o}j
ker T0 Vo such that Now is ainecly SuPPose inde penden T is Lineariy independe nf ButTU AvEkez T lineariy indePendent. contad
Nou is ineary Tun TVkt claim de Pendent amm) TQKtVk+1 +. 2 aiv U is basis tor ker T basis to v Tumis L. T is basis toIm (T).
{Tv1 Now Co verseiy let Ti Tv SPan W 1 Let we w TCai ET Tis Lineas TV here iニ) Tv T is surjec tive the Poved S0rution Hence
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Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) P...
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