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.
$\\alpha(\\beta v) =(\\alpha \\beta)v\\text{;}$
$(\\alpha + \\beta)v =\\alpha v + \\beta v\\text{;}$
$\\alpha(u + v) = \\alpha u + \\alpha v\\text{;}$
$1v=v\\text{;}$
where $\\alpha, \\beta \\in F$ and $u, v \\in V\\text{.}$
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 ${\\mathbf 0}\\text{,}$ 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 ${\\mathbb R}^n\\text{,}$ form a vector space over ${\\mathbb R}\\text{.}$ Given vectors $u = (u_1, \\ldots, u_n)$ and $v = (v_1, \\ldots, v_n)$ in ${\\mathbb R}^n$ and $\\alpha$ in ${\\mathbb R}\\text{,}$ we can define vector addition by
\n\\begin{equation*}\nu + v = (u_1, \\ldots, u_n) + (v_1, \\ldots, v_n) = (u_1 + v_1, \\ldots, u_n + v_n)\n\\end{equation*}\n
and scalar multiplication by
\n\\begin{equation*}\n\\alpha u = \\alpha(u_1, \\ldots, u_n)= (\\alpha u_1, \\ldots, \\alpha u_n).\n\\end{equation*}\n
Example20.2
If $F$ is a field, then $F[x]$ is a vector space over $F\\text{.}$ The vectors in $F[x]$ are simply polynomials, and vector addition is just polynomial addition. If $\\alpha \\in F$ and $p(x) \\in F[x]\\text{,}$ then scalar multiplication is defined by $\\alpha p(x)\\text{.}$
Example20.3
The set of all continuous real-valued functions on a closed interval $[a,b]$ is a vector space over ${\\mathbb R}\\text{.}$ If $f(x)$ and $g(x)$ are continuous on $[a, b]\\text{,}$ then $(f+g)(x)$ is defined to be $f(x) + g(x)\\text{.}$ Scalar multiplication is defined by $(\\alpha f)(x) = \\alpha f(x)$ for $\\alpha \\in {\\mathbb R}\\text{.}$ For example, if $f(x) = \\sin x$ and $g(x)= x^2\\text{,}$ then $(2f + 5g)(x) =2 \\sin x + 5 x^2\\text{.}$
Example20.4
Let $V = {\\mathbb Q}(\\sqrt{2}\\, ) = \\{ a + b \\sqrt{2} : a, b \\in {\\mathbb Q } \\}\\text{.}$ Then $V$ is a vector space over ${\\mathbb Q}\\text{.}$ If $u = a + b \\sqrt{2}$ and $v = c + d \\sqrt{2}\\text{,}$ then $u + v = (a + c) + (b + d ) \\sqrt{2}$ is again in $V\\text{.}$ Also, for $\\alpha \\in {\\mathbb Q}\\text{,}$ $\\alpha v$ is in $V\\text{.}$ We will leave it as an exercise to verify that all of the vector space axioms hold for $V\\text{.}$
Proposition20.5
Let $V$ be a vector space over $F\\text{.}$ Then each of the following statements is true.
$0v ={\\mathbf 0}$ for all $v \\in V\\text{.}$
$\\alpha {\\mathbf 0} = {\\mathbf 0}$ for all $\\alpha \\in F\\text{.}$
If $\\alpha v = {\\mathbf 0}\\text{,}$ then either $\\alpha = 0$ or $v = {\\mathbf 0}\\text{.}$
$(-1) v = -v$ for all $v \\in V\\text{.}$
$-(\\alpha v) = (-\\alpha)v = \\alpha(-v)$ for all $\\alpha \\in F$ and all $v \\in V\\text{.}$
Proof
To prove (1), observe that
\n\\begin{equation*}\n0 v = (0 + 0)v = 0v + 0v;\n\\end{equation*}\n
consequently, ${\\mathbf 0} + 0 v = 0v + 0v\\text{.}$ Since $V$ is an abelian group, ${\\mathbf 0} = 0v\\text{.}$
The proof of (2) is almost identical to the proof of (1). For (3), we are done if $\\alpha = 0\\text{.}$ Suppose that $\\alpha \\neq 0\\text{.}$ Multiplying both sides of $\\alpha v = {\\mathbf 0}$ by $1/ \\alpha\\text{,}$ we have $v = {\\mathbf 0}\\text{.}$
To show (4), observe that
\n\\begin{equation*}\nv + (-1)v = 1v + (-1)v = (1-1)v = 0v = {\\mathbf 0},\n\\end{equation*}\n
and so $-v = (-1)v\\text{.}$ We will leave the proof of (5) as an exercise.