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# Section20.1Definitions and Examples¶ permalink

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 $V$.