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Sagot :
The given set of functions are not linearly independent.
Given,
[tex]f_{1} (x) = x\\f_{2} (x) = x^{2} \\f_{3} (x) = 6x-2x^{2}[/tex]
We need,
[tex]c_{1} f_{1} (x)+c_{2} f_{2} (x)+c_{3} f_{3}(x)=0[/tex]
Substituting the values in equation we get,
[tex]c_{1} x+c_{2} x^{2} +c_{3} (6x-2x^{2} )=0\\[/tex]
Computing the equation we get,
[tex]c_{1} x+c_{2} x^{2} +c_{3} 6x-c_{3} 2x^{2}=0[/tex]
[tex](c_{1} +6c_{3} )x+(c_{2} -2c_{3} x^{2} =0[/tex]
This resolves to two equations
[tex](c_{1} +6c_{3})x =0\\(c_{2} -2c_{3} )x^{2} =0[/tex]
These will have an infinite set of solutions:
[tex]c_{1} =-6c_{3} \\c_{2} =2c_{3}[/tex]
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:https://brainly.com/question/18331568
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