Vector (source code)
= Vector
{title2=$v\in V$}
A <vector> is an element of a <vector space>. Its defining operations are <vector addition> and <scalar multiplication>. An arrow in a <Euclidean space> is one example; a <polynomial> or a <function> can also be a <vector> when the chosen <vector space> uses the corresponding addition and <scalar multiplication>.
For instance, the <polynomials> $p(x)=1+x$ and $q(x)=x-x^2$ belong to the real <vector space> of <polynomials> of degree at most two. Their <vector addition> gives $(p+q)(x)=1+2x-x^2$, and <scalar multiplication> gives $(2p)(x)=2+2x$. In the <monomial basis> $(1,x,x^2)$ these are the <coordinate vectors> $(1,1,0)$, $(0,1,-1)$, $(1,2,-1)$ and $(2,2,0)$ respectively. The <vectors> are the <polynomials>; these tuples are their representations in a chosen <basis>.
A general <vector space> does not itself prescribe lengths or <angles>. A <norm> supplies a size, while an <inner product> also supplies angle information and <orthogonality>. The familiar arrow geometry below belongs to <Euclidean vectors>, where this extra structure is available.
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