Final answer:
The set of polynomials 1, x, x², ..., xⁿ is a basis for the vector space V of polynomials with degree at most n because it is both linearly independent and spans V, satisfying the conditions for a set to be a basis.
Step-by-step explanation:
To show that the set of polynomials 1, x, x², ..., xⁿ is a basis of the vector space V consisting of polynomials of degree at most n, we need to demonstrate two properties: that the set is linearly independent and that it spans the vector space V.
First, we consider linear independence. Suppose there is a linear combination of these polynomials that equals zero: a₀ + a₁x + a₂x² + ... + aⁿxⁿ = 0, where a₀, ..., aⁿ are coefficients. For this equation to hold for all values of x, each coefficient must be zero. This implies that no polynomial in this set can be written as a linear combination of the others, proving linear independence.
To prove that the set spans V, we note that any polynomial in V can be expressed as p(x) = b₀ + b₁x + b₂x² + ... + bⁿxⁿ, where b₀, ..., bⁿ are coefficients. Thereby, any polynomial of degree at most n in V can be written as a combination of the set 1, x, x², ..., xⁿ, meaning the set spans V.
Therefore, as the set 1, x, x², ..., xⁿ is both linearly independent and spans V, it is a basis for the vector space of polynomials of degree at most n.