Question

Let y be a differentiable function such that y' – 7y = 7, where y' is the first derivative of y. Define addition yı y2 = yı + y2 + 1 and scalar multiplication c Oyı = cyı + c – 1. Let V be the set of all differentiable functions that solve y' – 7y = 7. Note that these functions fit the form y = ket – 1. Determine if V is a vector space.

Answer 1

+ the elements in v are in ten form ya ketti cet vecter addition defined by S EN & Yoy = 4 ty ti and scalar multiplication defined by choy = y + (-1 Addition properties! ☺ y, a kiel akzeil »y, ay = ket+keit 3 (kitkzletfi EV i is closure w for 9, 194 Ev we must have (4@4x)Ⓡyz? YBh@y) 16 for kao, y = -1 EV and for any yake - imply ye@ y = [-j@fcetti) = 4+ ka then take this sy Ye=eV is an Identity element in V. Zeno vector) th akery EV thun yanken GV such that a Invesse of y, is % 4,642= ye y-te x an inverse of kett, is -kether. 6 ty iGEV, Y 842= %@y, Crit) is an abelian puupa

Date Page No. Eastong markiplication Pauperties y Evi Cefi ea cyte-1 o C (kot bi g tey ☺ =cket Ev. 2- = cket in for CE FI YEV > COYEV 1 and cucer (etc.) oy = Center) y + catre til (altre) (nett,] Acite) al = litre)k ethit @ (coy) ② (coy) = ayta-1tay +(2-) =(ittely + ethan = Citre) (teett, teatteri = (steel kett (utc) tk) - [= Catez) ke? I J from @ +6 > (1+2) Oya (9.9) (C209)

@ By ofaking scalar meltiplication property. effel CEF, CO 69@yz) = co Cynty,+l) = c Cynthe+1)+cal = cy, tcy,+22-1 and (coy) (coyz) = (cy, tc-1) (cyztery) = cy,+c-1 tCyntc- + = cyty 2+26-1 cogioy) = (604) ② (com) @ alber, Casley = aby+ ab- and y) = a0(bytbl) = a(sytbil) tal - asytabaqlay = aby 295-1 li (as)oy= aoboy) 0 .EF. lng= lry +1=1=y LEF, 1.g=y FYEN v bean vector space. 1