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 $v^T.$

| | | | | created: 2018-04-09 23:38:57 | modified: 2020-11-29 14:13:45 | by: bookofproofs | references: [7937]