Question

15. (5 points) Enough of matrices, now let us consider the vector space P2. Let P1 = 2 – x2, P2 = 3x, and P3 = x2 + x – 2, determine whether the polynomials above are linearly independent or dependent in P2. Use just the definition, nothing fancy.

Answer 1

stway Lexul QISE let ou consider the Vector space Pa. let P.2-?, P2= 3x and P2 = x²+x-2 ? To check linearly independent or lineasly dependent. wewe Basic definition i. I hel V be a vector space ever field five VCF) & V., V2, ? I be an n-vectors then there vectors are l.Torld if a suppose I, 4, an Eflfield) I comept e divto, vt anvn D y Eli Vi=0 we 5 dico (all in ton we to lie all di aue zero then these vector are l I dolue L o therwise if any one di is non-zero then those question vector alle linearly dependent. o e Sol:- Here P [x] is a vector space over field Blie woefficient aue borr R (Real number)_4 let L., 2 , 4 Aue three scalay from field R. To check for lit or L.D we use ie Lip + 22 h tel3pz = 0 + 0x + 0x2 » 2, (2-2²) to (3x) + 23 (x²+1-2) = 0 ton +012 = (-dtds)x? + (342 +d3] je +20,-24y = 0 70x70x2 - 0 - -1, +4₂=0 t 1 342 743 =0 Y on compaining Both are same h 21, -28₂=0 TL. HS = RHS fom 60, equations au : 24-0, % -2=0 (2) 342-43 =0

- Solving eq - we get one free rasiable Lg, so d = 43, & 2 = 43/3 ! so all li are not hero , hence the p, (e), p 2 (x) & P₃(e) are linegsly dependent vector in Pelu] way It is clearly see that 3(p. +P3 ) = P2 i.€ 34,+) = P2 - 3(2-x2+x²+2-2)= 3x - 3x = 3x. flence B (80) is linearly dependent on linear Combination of P & P 2 (Hence proved
these vectors are Linearly dependent. I written in two ways you can easily undery the concept.

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