Can a list of 2016 polynomials span P2016 Explain why or why

Can a list of 2016 polynomials span P_2016? Explain why or why not. (b) Can a list of 2018 polynomials be linearly independent in P_2016? Explain why or why not.

Solution

a) P 2016 would be of the form

a0+a1x+a2x^2+...+a₂₀₁₆x²⁰¹⁶

Thus there are 2011 coefficients for x including coefficient for x power 0

i.e. there are 2017 coefficients used here to write a P2016.

Hence dim P2016 = 2017

So 2016 polynomials cannot span P2016

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b) P2016 is of dimension 2017

Hence the minimum number of linearly independent vectors in P2016 = 2017

Any number greater than 2017 also can be linearly independent.

Hence 2018 polynomials can be linearly independent in P2016