We are now ready to define vector spaces. The idea is to observe that sets of column vectors, or row vectors, or more generally matrices of a given size, all come equipped with a notion of addition and scalar multiplication and all obey the same collection of simple algebraic rules, for example, that addition is commutative, that scalar multiplication distributes over vector addition, and so on. We will define a vector space as any set with operations of addition and scalar multiplication obeying similar rules to those satisfied by column vectors. The power of doing this is that it lets us apply our theory in seemingly entirely different contexts.
Let $\mathbb{F}$ be a field. An $\mathbb{F}$-vector space is a set $V$ with
a special element ${\text{\U0001d7ce}}_{V}$ called the zero vector
an operation + called addition
a way to multiply elements of $V$ by elements of $\mathbb{F}$, called scalar multiplication
such that for all $\text{\mathbf{u}},\text{\mathbf{v}},\text{\mathbf{w}}$ in $V$ and all $\lambda ,\mu $ in $\mathbb{F}$,
$\text{\mathbf{v}}+\text{\mathbf{w}}=\text{\mathbf{w}}+\text{\mathbf{v}}$
$\text{\mathbf{u}}+(\text{\mathbf{v}}+\text{\mathbf{w}})=(\text{\mathbf{u}}+\text{\mathbf{v}})+\text{\mathbf{w}}$
${\text{\U0001d7ce}}_{V}+\text{\mathbf{v}}=\text{\mathbf{v}}$
there exists $\text{\mathbf{x}}\in V$ such that $\text{\mathbf{x}}+\text{\mathbf{v}}={\text{\U0001d7ce}}_{V}$
$\lambda (\mu \text{\mathbf{v}})=(\lambda \mu )\text{\mathbf{v}}$
$1\text{\mathbf{v}}=\text{\mathbf{v}}$
$\lambda (\text{\mathbf{v}}+\text{\mathbf{w}})=\lambda \text{\mathbf{v}}+\lambda \text{\mathbf{w}}$
$(\lambda +\mu )\text{\mathbf{v}}=\lambda \text{\mathbf{v}}+\mu \text{\mathbf{v}}$
You sometimes see the phrase “vector space over $\mathbb{F}$”, which means the same thing as $\mathbb{F}$-vector space.
The elements of vector spaces can be anything at all. They don’t have to look like column or row vectors. Here are some examples of vector spaces.
${\mathbb{R}}^{n}$ is a real vector space, ${\u2102}^{n}$ is a complex vector space, and if $\mathbb{F}$ is any field then ${\mathbb{F}}^{n}$, the set of all height $n$ column vectors with entries from $\mathbb{F}$ is an $\mathbb{F}$-vector space.
${M}_{m\times n}(\mathbb{R})$, the set of all $m\times n$ matrices with real entries, is a real vector space with the zero vector being the all-zeroes matrix. Similarly for any other field.
$\{0\}$ with the only possible operations is an $\mathbb{F}$-vector space, for any field $\mathbb{F}$, the zero vector space.
Let $\mathcal{F}$ be the set of all functions $\mathbb{R}\to \mathbb{R}$. Define $f+g$ to be the function $\mathbb{R}\to \mathbb{R}$ given by $(f+g)(x)=f(x)+g(x)$ and, for a real number $\lambda $ and a function $f$, define $\lambda f$ by $(\lambda f)(x)=\lambda f(x)$. Then $\mathcal{F}$ is a real vector space with the zero vector being the constant function taking the value 0.
If $A$ is a $m\times n$ matrix, the set of all solutions of $A\text{\mathbf{x}}=0$ is a vector space. This is the nullspace $N(A)$ we met in Definition 3.6.3.
The set of all real solutions to the differential equation ${y}^{\prime \prime}+a{y}^{\prime}+by=0$ is a vector space, with the definitions of addition and scalar multiplication as in $\mathcal{F}$ above.
The set $\mathbb{F}[x]$ of all polynomials in one variable x is a $\mathbb{F}$-vector space, as is the set ${\mathbb{F}}_{\u2a7dn}[x]$ of all polynomials in $x$ of degree at most $n$.
the set of magic matrices, those whose row sums and column sums are all equal, is a vector space with the usual matrix scalar addition and multiplication.