Question

Q10.2 3 Points Let V and W be finite dimensional vector spaces over R and T:V + W be linear. Let Vo be a subspace of V and Wo = T(V). (Select ALL that are TRUE) If T is surjective then Vo = {v E V: there is w E Wo such that T(v) = w}. If T is injective then dim(V.) = dim(Wo). dim(ker(T) n ) = dim(V.) - dim(Wo). Save Answer

Answer 1

Let and T! V - Way and Wo 'T T be We see that v and w be finite dimensional rector spaces over IR be linear. Let to be a subspace of v T(4) Let Surjective 0 det U Evo > T(u) ETC VO) = Wo = T(4) for some w two. Since Vo sv V: There it wt. Wo such that T(2) = w} {rev: there is weWo such that Now we show that 7 x 6 T(W) =W} & to Chose a non - Zemo veeter x E Kert ivo, tren T(*) on two count So, whenever ker T& Yo 3 a non-zero veetor REVIVO sit TGEWO Vos {vev! Let Ker T Hence The set Voo is a proper subset of {rev: Such that T(%) = w} rtv : there is weWo Such that The). T W Ve Kort Wo So, (1) is not true (2 Let ker Let baon } R we show that be = Wo let wt Wo= then 3 vt vo Sit. - w Since {v., So, r=Cinet = T be injective, then {ov? {v, vz, .vn] be of Vo {Tron), Tha), , Thou will a basin of T(vo) T(Vo) T (2) {V.,---, *m} j a barab of vo. it en ren for some case, GER = T (2) C, TV,) + +GT (2) GT (0) + + (nT (2) So Wo i generated by the set {rto.), ttu)- 2,76.] we show that {rfur), T), ...,T(2n} in linearly independent in w. CT(12) + + Cm T (2) => T (cirst + Cn 29 ) + Chont Kert = {e} Grit + Cn vn = Oy 2n} linearly independent set of Vectors in v =(n=0 which shows that {T(+1), (2) basiſ of T(vo) dim (wo) Now let =tw c, vt Since {Q, ... ור G = = T(20m) in a - Wo so, dim Vo n = Hence (2) is true. (3) We ) so a hub Hence {vis a basis Since T(29) འ Know That kert is a subspace of v. So. Kertn Vo is a subspace of v Let {re, ... , ve} be a basis of Kertvo Since KerTn Vo svo Ker inyo is o spe ce of Vo. {V, --, V} Vx} can be extended to Vk, Vario -, zens of y, says So, dim (Vo) = n and dhim (ker TOVO) = K. TV-) T(2x) = ow, so { by (2) Wo is generated by {t(vx+1), -> T(200)} Ek+iT (2+1) + + Cn T (on) = Ow => T(Carl Varit + C vn ) = ow 7 Ckti vutit + (n un E Kert Chti Vrtit +62 C Vo , [": { 1, -, Ver... on} is a basis of vo ]. CR + lektit + Cnr Ekoran Vo Since {ves... va} is a basis of Ker In Vo, torn=diret + dk Vk let Abo Hence Chei Val + => d, vt + die vra Since {en Cat Vati v.} is a lineanly independent subset of d, - Crete So, { T (Akte), --. T(on)} is a linearly independent generating subset of wo and hence a basis of wo then Hence din Wo n-k So, drus Vo din Wo (n-k) intk din (kertnyo) So, () is true Scanned with CamScanner