Website owner:  James Miller

                                   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 + ... + a_nxⁿ 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 + ... + a_nxⁿ of degree n in one variable form a finite dimensional vector space whose dimension is n+1. Why? Because every polynomial whose degree is 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 n .