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                                   Space of polynomials                            

The set of all polynomials (of all degrees) in one variable. The set of all polynomials in one variable a0 + a1x + a2x2 + ... + anxn form a vector space. Why? Using the usual definition of the operations of addition and of multiplication by a number for polynomials they satisfy the eight postulates for a linear space. This space is infinite dimensional since the vectors 1, x, x2, ... , xn are linearly independent for any n.

The set of all polynomials of degree ≤ n in one variable. The set of all polynomials a0 + a1x + a2x2 + ... + anxn of degree ole.gif n in one variable form a finite dimensional vector space whose dimension is n+1. Why? Because every polynomial whose degree is ole1.gif n is a linear combination of the n+1 linearly independent vectors 1, x, x2, ... , xn. These n + 1 vectors constitute a basis for the set of all polynomials in one variable of degree ole2.gif n . 


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