While most of the examples and applications we shall consider are vector spaces over the field of real or complex numbers, for the examples below, we let \(F\) denote any field. First recall the definition of a vector space [click the link to toggle the definition].
For an integer \(n \ge 1\text{,}\)\(V=F^n\text{,}\) the set of \(n\)-tuples of numbers in \(F\) viewed as column vectors, is a vector space over \(F\text{.}\)
For integers \(m,n \ge 1\text{,}\) we have the vector space of \(m\times n\) matrices, denoted \(M_{m\times
n}(F).\) Column vectors are the matrices in \(M_{m\times1}(F)\text{,}\) while row vectors are matrices in \(M_{1\times n}(F)\text{.}\)
For an integer \(n \ge 1\text{,}\) we denote by \(P_n(F)\) the vector space of polynomials of degree at most \(n\) having coefficients in \(F\text{.}\)
The vector space of all polynomials with coefficients in \(F\) is often denoted as \(P(F)\) in many linear algebra texts, though in more advanced courses (say abstract algebra) the more typical notation is \(F[x]\text{,}\) a notation we shall use here.
