Determine whether the given set of functions is linearly dependent or linearly i

Brittney Lord 2021-09-15 Answered

Determine whether the given set of functions is linearly dependent or linearly independent on the interval (,)
f1(x)=1+x,

f2(x)=x,

f3(x)=x2

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gotovub
Answered 2021-09-16 Author has 98 answers

From Theorem we know that the function f1,f2, and f3 are linearly independent if and only if W(f1,f2,f3)0
Where W(f1,f2,f3)=|f1f2f3f1f2f3f1f2f3|
f1(x)=1+x f2(x)=x f3(x)=x2
W(f1,f2,f3)=|1+xxx2112x002|
Using the properties of determinants we have
W(f1,f2,f3)=(1+x)|12x02|x|12x02|+x2|1100|
W(f1,f2,f3)=(1+x)(20)x(20)+x2(00)
W(f1,f2,f3)=(1+x)(2)x(2)+x2(0)
W(f1,f2,f3)=2+2x2x+0W(f1,f2,f3)=2
W(f1,f2,f3)0
Thus, the given function are linearly independent
Result: The given functions are linearly independent As their Wronskian W(f1,f2,f3)=20

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