1. The problem is to understand polynomials in vector spaces. 2. A polynomial in a vector space is an expression of the form $$p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$ where the coefficients $a_i$ belong to a field (like real numbers) and $x$ is a vector variable. 3. Important rules: - Vector spaces allow addition and scalar multiplication. - Polynomials form a vector space because sums and scalar multiples of polynomials are also polynomials. 4. For example, if $p(x) = 2 + 3x$ and $q(x) = 1 - x$, then their sum is $$p(x) + q(x) = (2+1) + (3-1)x = 3 + 2x$$ which is also a polynomial. 5. Scalar multiplication by $c$ of $p(x)$ is $$c p(x) = c(2 + 3x) = 2c + 3c x$$ which is also a polynomial. 6. Thus, polynomials with coefficients in a field form a vector space under usual addition and scalar multiplication. 7. This vector space is infinite dimensional because polynomials can have arbitrarily high degree. 8. The set of monomials $\{1, x, x^2, x^3, \ldots\}$ forms a basis for this vector space. 9. Each polynomial can be uniquely expressed as a finite linear combination of these basis vectors. 10. This understanding is fundamental in algebra and functional analysis.