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1.prove that the following three functions are linearly dependent

f1=x^2; f2(x)=1-x^2; f3=2+x^2

User Szydan
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1 Answer

6 votes

Proof:

Given any functions
f_(1)(x),f_(2)(x),f_(3)(x) they are linearly dependent if we can find values of
c_(1),c_(2),c_(3) such that


c_(1)f_(1)(x)+c_2f_(2)(x)+c_(3)f_(3)(x)=0

Using the given functions in the above equation we get


c_(1)f_(1)(x)+c_2f_(2)(x)+c_(3)f_(3)(x)=0\\\\c_(1)x^(2)+c_(2)(1-x^(2))+c_(3)(2+x^(2))=0\\\\\Rightarrow (c_(1)-c_(2)+c_(3))x^(2)+c_(1)+c_(2)+2c_(3)=0

This will be satisfied if and only if


c_1-c_2+c_3=0,c_1+c_2+2c_3=0

Solving the equations we get


c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0

Since we have 3 variables and 2 equations thus we will get many solutions

one being if we put
c_3=1 we get


c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0\\\\c_1=(-3)/(2)\\\\c_2=(-1)/(2)

Thus we have
c_1=(-3)/(2),c_2=(-1)/(2),c_3=1 as one solution. Hence the given functions are linearly dependent.

User Talhature
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