Question

Determine whether the given set of functions is linearly independent on the interval (−[infinity], [infinity]). f1(x) = x, f2(x) = x2, f3(x) = 6x − 2x2 linearly dependent linearly independent

Answer 1

The given set of functions are not linearly independent.

Given,

$f_{1} (x) = x\\f_{2} (x) = x^{2} \\f_{3} (x) = 6x-2x^{2}$

We need,

$c_{1} f_{1} (x)+c_{2} f_{2} (x)+c_{3} f_{3}(x)=0$

Substituting the values in equation we get,

$c_{1} x+c_{2} x^{2} +c_{3} (6x-2x^{2} )=0$

Computing the equation we get,

$c_{1} x+c_{2} x^{2} +c_{3} 6x-c_{3} 2x^{2}=0$

$(c_{1} +6c_{3} )x+(c_{2} -2c_{3} x^{2} =0$

This resolves to two equations

$(c_{1} +6c_{3})x =0\\(c_{2} -2c_{3} )x^{2} =0$

These will have an infinite set of solutions:

$c_{1} =-6c_{3} \\c_{2} =2c_{3}$

Two functions are said to be linearly independent if neither function is a constant multiple of the other.

Here, it is clear that the given functions are not linearly independent.

Learn more about linearly dependent or independent functions here:brainly.com/question/18331568

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