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Section20.1Definitions and Examples

A vector space \(V\) over a field \(F\) is an abelian group with a scalar product \(\alpha \cdot v\) or \(\alpha v\) defined for all \(\alpha \in F\) and all \(v \in V\) satisfying the following axioms.

  • \(\alpha(\beta v) =(\alpha \beta)v\);

  • \((\alpha + \beta)v =\alpha v + \beta v\);

  • \(\alpha(u + v) = \alpha u + \alpha v\);

  • \(1v=v\);

where \(\alpha, \beta \in F\) and \(u, v \in V\).

The elements of \(V\) are called vectors; the elements of \(F\) are called scalars. It is important to notice that in most cases two vectors cannot be multiplied. In general, it is only possible to multiply a vector with a scalar. To differentiate between the scalar zero and the vector zero, we will write them as 0 and \({\mathbf 0}\), respectively.

Let us examine several examples of vector spaces. Some of them will be quite familiar; others will seem less so.

Example20.1

The \(n\)-tuples of real numbers, denoted by \({\mathbb R}^n\), form a vector space over \({\mathbb R}\). Given vectors \(u = (u_1, \ldots, u_n)\) and \(v = (v_1, \ldots, v_n)\) in \({\mathbb R}^n\) and \(\alpha\) in \({\mathbb R}\), we can define vector addition by \begin{equation*}u + v = (u_1, \ldots, u_n) + (v_1, \ldots, v_n) = (u_1 + v_1, \ldots, u_n + v_n)\end{equation*} and scalar multiplication by \begin{equation*}\alpha u = \alpha(u_1, \ldots, u_n)= (\alpha u_1, \ldots, \alpha u_n).\end{equation*}

Example20.2

If \(F\) is a field, then \(F[x]\) is a vector space over \(F\). The vectors in \(F[x]\) are simply polynomials, and vector addition is just polynomial addition. If \(\alpha \in F\) and \(p(x) \in F[x]\), then scalar multiplication is defined by \(\alpha p(x)\).

Example20.3

The set of all continuous real-valued functions on a closed interval \([a,b]\) is a vector space over \({\mathbb R}\). If \(f(x)\) and \(g(x)\) are continuous on \([a, b]\), then \((f+g)(x)\) is defined to be \(f(x) + g(x)\). Scalar multiplication is defined by \((\alpha f)(x) = \alpha f(x)\) for \(\alpha \in {\mathbb R}\). For example, if \(f(x) = \sin x\) and \(g(x)= x^2\), then \((2f + 5g)(x) =2 \sin x + 5 x^2\).

Example20.4

Let \(V = {\mathbb Q}(\sqrt{2}\, ) = \{ a + b \sqrt{2} : a, b \in {\mathbb Q } \}\). Then \(V\) is a vector space over \({\mathbb Q}\). If \(u = a + b \sqrt{2}\) and \(v = c + d \sqrt{2}\), then \(u + v = (a + c) + (b + d ) \sqrt{2}\) is again in \(V\). Also, for \(\alpha \in {\mathbb Q}\), \(\alpha v\) is in \(V\). We will leave it as an exercise to verify that all of the vector space axioms hold for \(V\).