Exercises 6.111.Lety1=x−1,y2=x1/2, andy3=x. We want to find constantsc1,c2, andc3such thatc1y1+c2y2+c3y3=c1x−1+c2x1/2+c3x= 0,for allxon the interval (0,∞). This equation must hold ifx= 1, 4, or 9 (or any other valuesforxin the interval (0,∞)). By plugging these values forxinto the equation above, we seethatc1,c2, andc3must satisfy the three equationsc1+c2+c3= 0,c14+ 2c2+ 4c3= 0,c19+ 3c2+ 9c3= 0.Solving these three equations simultaneously yieldsc1=c2=c3= 0. Thus, the only way forc1x−1+c2x1/2+c3x= 0 for allxon the interval (0,∞), is forc1=c2=c3= 0. Therefore,these three functions are linearly independent on (0,∞).13.A linear combination,c1x+c2x2+c3x3+c4x4, is a polynomial of degree at most four, and so,by the fundamental theorem of algebra, it cannot have more than four zeros unless it is thezero polynomial (that is, it has all zero coeﬃcients). Thus, if this linear combination vanishesonan interval, thenc1=c2=c3=c4= 0. Therefore, the functionsx,x2,x3, andx4arelinearly independent on any interval, in particular, on (
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