list vector

Let $𝕂$ be a field and $n$ a positive natural number. We define${𝕂}^n$ to be the set of all mappings from the index list$(1,2,\mathrm{\dots },n)$ to $𝕂$. Such a mapping $a\in {𝕂}^n$ isjust a formal way of speaking of a list of field elements$a^1,\mathrm{\dots },a^n\in 𝕂$.

The above description is somewhat restrictive. A more flexibledefinition of a list vector is the following. Let $I$ be a finitelist of indices, $I=(1,\mathrm{\dots },n)$ is onesuch possibility, and let ${𝕂}^I$ denote the set of all mappingsfrom $I$ to $𝕂$. A list vector, an element of ${𝕂}^I$, isjust such a mapping. Conventionally, superscripts are used to denotethe values of a list vector, i.e. for $u\in {𝕂}^I$ and $i\in I$,we write $u^i$ instead of $u(i)$.

We add and scale list vectors point-wise, i.e. for $u,v\in {𝕂}^I$ and $k\in 𝕂$, we define $u+v\in {𝕂}^I$ and $ku\in {𝕂}^I$, respectively by

${(u+v)}^i$$=u^i+v^i,i\in I,$

${(ku)}^i$$=k{u}^i,i\in I.$

We also have the zero vector $\mathrm{𝟎}\in {𝕂}^I$, namely the constant mapping

$${\mathrm{𝟎}}^i=0,i\in I.$$

The above operations give ${𝕂}^I$ thestructure of an (abstract) vector space over $𝕂$.

Long-standing traditions of linear algebra hold that elements of${𝕂}^I$ be regarded as column vectors. For example, we write$a\in {𝕂}^n$ as

$$a=\left(\begin{array}{c}\hfill a^1\hfill \\ \hfill a^2\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill a^n\hfill \end{array}\right).$$

Row vectors are usually taken to represents linear forms on${𝕂}^I$. In other words, row vectors are elements of the dualspace ${\left({𝕂}^I\right)}^{*}$. The components of a row vector arecustomarily written with subscripts, rather than superscripts. Thus,we express a row vector $\alpha \in {\left({𝕂}^n\right)}^{*}$ as

$$\alpha =({\alpha }_1,\mathrm{\dots },{\alpha }_n).$$