1.prove that the following three functions are linearly dependent
f1=x^2; f2(x)=1-x^2; f3=2+x^2

Proof:

Given any functions they are linearly dependent if we can find values of such that

Using the given functions in the above equation we get

This will be satisfied if and only if

Solving the equations we get

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

one being if we put we get

Thus we have as one solution. Hence the given functions are linearly dependent.

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