By Katz N.M.

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**Additional info for A conjecture in arithmetic theory of differential equations (Bull. Soc. Math. Fr. 1982)**

**Example text**

3 a a or, 2 (y - 1) 3 + 3 (y - 1) 2 y y _ (y - 1) _ 1 = 0 y or, 2(y - 1)3 + 3(y - 1)2y - (y - 1)y2 - y3 = 0 or, 3y 3 (vi) Let y = lly2 - + 9y - 2 = O. a~ a~"( 3 3 or, y = - - = - or, "( = -. "( "( Y Also "( is a root of the given equation . '. "(3 - 2"(2 or, (~)3 _ 2(~)2 + (~) _ 3 = 0 y y Y 9 6 or - - 'y3 y2 1 + -Y - 1= 0 + 6y - 9 = O. + ~ or, Y = a + ~ + "( - "( = p - or, y3 - y2 (vii) Let y = a "( or, "( Since "( is a root of the given equation, "(3 - p"(2 + q"( _ r = 0 or, (p~y)3_p(p_y)2+q(p-y)_r=0 or, y3 - 2py2 + (p2 +q)y - (pq - r) = O.

T. addition and scalar multiplication. Theorem_1 A non-emp~ ~ubset W of V is a subspace of V iff (i) a, j3 E W =} a + j3 EE Wand (ii) a E W, e E F =} cal E W. Ex. 1 (i) Let V = {(x, Y/' z) : x, y, z E R} and if W = {(x, y, z) : x - 3y + 4z = O} then prove t~at W is a subspace of V. I (ii) Let V = {(x,y,z) : i,y,z E R} and if W = {(x,y,z) : x2 then prove that W is not a subspace of V . • + y2 = z2} SOLUTION: (i) Let a = (Xl, yl, zI) and Then Xl - 3Yl + 4Zl 7J = (X2' Y2, Z2) be two vectors of W. = 0 and X2 - 3Y2 + 4Z2 = o.

4 Linearly independent: A set of vectors {Xl, X2, ... , xn} of En is said to be linearly independent if the only set of Ci for which CIXI holds, be Ci + C2X2 + ... + CnXn = 0 = 0, i = 1,2, ... , n. Q. 1 Define linearly dependent and independent vectors. bl al CJ a2 b2 C2 . a3 b3 C3 If Ll = 0 then the vectors (al,bl,cl), (a2,b 2,c2) and (a3,b3,c3) are linearly dependent and if Ll =f. 0 then they are linearly independent. Formula 2 Let Ll = Ex. 2 (i) Show that the vectors (1,0,0)' (0,1,1), (1,1,1) in the real vector space ll3 are linearly dependent.