Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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
#SPJ4
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
