◀ ▲ ▶Branches / Algebra / Definition: Column Vectors and Row Vectors
Similar to the introduced matrices, in the following definition, we introduce the notion of a "vector" from a pure notational perspective as a special case of a matrix.
Definition: Column Vectors and Row Vectors
Let $n\ge 1$ be a natural number. A column vector (or just vector) with $n$ field elements is a matrix over a field $F$ with just one column and many rows, i.e. an element of $M_{m\times 1}(F):$
$$v=\pmatrix{\alpha_1\\\vdots\\\alpha_m}$$
Similarly, a row vector is transposed column vector
$$v^T=\pmatrix{\alpha_1,&\ldots&,\alpha_m}$$
A row vector is an element of $M_{1\times m}(F)$.
Notes
 Many sources introduce vectors differently, namely as elements of the Cartesian product $F^n$, where $F$ is a field.
 We prefer to consider vectors as special cases of matrices with a single column. This will allow us to derive many properties of vectors as special cases of properties of matrices. In particular, we gained already the notion of the transposed vector $v^T$ that cannot be derived from the Cartesian definition without additional explanation.
 Nevertheless, we will sometimes follow the convention and write $v\in F^n$ instead of $v\in M_{m\times 1}(F)$.
 Also by convention, we are going to denote vectors by small Latin or Greek letters, e.g. $v$, and we always mean a column vector by $v$ and row vectors by $v^T.$
Mentioned in:
Chapters: 1
Corollaries: 2
Definitions: 3 4 5
Proofs: 6
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Knabner, P; Barth, W.: "Lineare Algebra  Grundlagen und Anwendungen", Springer Spektrum, 2013