The functions f_1, f_2, ..., f_m defined on the interval I are said t.pdf

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The functions f_1, f_2, ..., f_m defined on the interval I are said to be linearly dependent on I if at least one of them can be expressed as a linear combination of the others on I. Equivalently, they are linearly dependent on I if there exist constants c_1, c_2, ..., c_m, not all zero, such that c_1 f_1 (x) + c_2 f_2 (x) + middot middot middot + c_m f_m (x) = 0 for all x I. Otherwise, they are said to be linearly independent on I. Show that the functions f_1 (x) = e^x f_2 (x) = e^-2x f_3 (x) = 3e^x - 2e^-2x are linearly dependent on (-infinity, infinity). Solution Given that f1(x) = ex , f2(x) = e-2x and f3(x) = 3ex - 2e-2x Then there exists non-zero constants C1 = - 3 , C2 = 2 and C3 = 1 such that C1f1(x) + C2 f2(x) + C1f3(x) = 0. Hence The functions f1( x) = ex , f2(x) = e-2x and f3(x) = 3ex - 2e-2x are linearly dependent for all x *( - , ).

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(a) Natural Numbers : N = {1,2,3,4,...} (b) Whole Numbers : W = {0,1,2,3,4, } (c) Integer Numbers :  or Z = {...–3,–2,–1, 0,1,2,3, }, Z+ = {1,2,3,....}, Z– = {–1,–2,–3, } Z0 = {± 1, ± 2, ± 3, } (d) Rational Numbers : p Q = { q ; p, q  z, q  0 } (i) R0 : all real numbers except 0 (Zero). (j) Imaginary Numbers : C = {i,, } (k) Prime Numbers : These are the natural numbers greater than 1 which is divisible by 1 and itself only, called prime numbers. Ex. 2,3,5,7,11,13,17,19,23,29,31,37,41,... (l) Even Numbers : E = {0,2,4,6, } (m) Odd Numbers : O = {1,3,5,7, } Ex. {1, Note : 5 , –10, 105, 3 22 20 7 , 3 , 0 ....} The set of the numbers between any two real numbers is called interval. (a) Close Interval : (i) In rational numbers the digits are repeated after decimal. (ii) 0 (zero) is a rational number. (e) Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q ,called irrational numbers Ex. { , ,21/3, 51/4,  ,e, } Note: (i) In irrational numbers, digits are not repeated after decimal. (ii)  and e are called special irrational quantities. (iii)  is neither a rational number nor a irrational number. (f) Real Numbers : {x, where x is rational and irrational number} 20 [a,b] = { x, a  x  b } (b) Open Interval: (a, b) or ]a, b[ = { x, a < x < b } (c) Semi open or semi close interval: [a,b[ or [a,b) = {x; a  x < b} ]a,b] or (a,b] = {x ; a < x  b} Let A and B be two given sets and if each element a  A is associated with a unique element b  B under a rule f , then this relation is called function. Here b, is called the image of a and a is called the pre- image of b under f. Note : (i) Every element of A should be associated with Ex. R = { 1,1000, 20/6,  , , –10, – ,.....} 3 B but vice-versa is not essential. (g) Positive Real Numbers: R+ = (0,) (h) Negative Real Numbers : R– = (– ,0) (ii) Every element of A should be associated with a unique (one and only one) element of but any element of B can have two or more rela- tions in A. 3.1 Representation of Function : It can be done by three methods : (a) By Mapping (b) By Algebraic Method (c) In the form of Ordered pairs (A) Mapping : It shows the graphical aspect of the relation of the elements of A with the elements of B . Ex. f1: f2 : f3 : f4 : In the above given mappings rule f1 and f2 shows a function because each element of A is associated with a unique element of B. Whereas f3 and f4 are not function because in f 3, element c is associated with two elements of B, and in f4 , b is not associated with any element of B, which do not follow the definition of function. In f2, c and d are associated with same element, still it obeys the rule of definition of function because it does not tell that every element of A should be associated with different elements of B. (B) Algebraic Method : It shows the relation between the elem

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