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W is a rele that A linear transformation T from a vector space V into a...

W is a rele that A linear transformation T from a vector space V into a vector space assigns to each vector 2 in V a unique v

W is a rele that A linear transformation T from a vector space V into a vector space assigns to each vector 2 in V a unique vector T() in W. such that (1) Tutu = Tu+Tv for all uv in V, and (2) Tſcu)=cT(u) for all u in V and all scalar c. *** The kernel of T = {UE V , T(U)=0} The range of T = {T(U) EW , ue V } Define T :P, - R K by T(m) = PO by 1 p = 0(1) a. Show that I is a linear transformation. b. Find The kernel of T and the range of T.
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Answer : Tip IR² defined by T(P) = (pro) P(1) where P2 ů a polynomial vector space & I of degree at most 2. het PEP2 impliesconsider T(P+q), - T(P+q) = T((aotain +Q₂x2)+ (bot bine + b2x2)) = T (aotbot (atb, ) x + (Az rb2) a2) aotbo | Clotbo tautbit(ii) het PEP₂ and cur any scalar un IR. Consider T(cp), T (cp) s TIC (Aotwm +Q, x)). = T( cao tea,x + ca, x²) I Cao caot catcI Kernel of T = { PEP2 a T(P) = 0 - 2 notara + a₂x² EP29 Pz ġ li to 10 Taotaite haot anta₂x² EP2 i varp 2 Co = and aotluta=08

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