3.1 Column Vector

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 3.1.1 Vertex Model of Points, 3.1.2 Vector Model of Mass-Spring Chain
Constructions: 4.4 Transpose of a Vector
Generalizations: 3.4 Row Vector

Properties: 3.2 The ith Coefficient of a Column Vector, 3.3 R^n, 3.5 Equal Column Vectors
Sufficiencies: N/A
Questions: N/A

Column Vector

Columns vectors are vertical arrays of real numbers. Let $n \in N$ . A column vector $\vec{x} \in R^{n \times 1}$ has $n$ real-valued entries, organized in $n$ rows and one column, in the form

\vec{x} = [\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}] .

For each $i \in {1, 2, \dots, n}$ , the entry in the $i$ th row of $\vec{x}$ , labeled $x_{i}$ , is a real number.

Remark. When writing $\vec{x} \in R^{n \times 1}$ , the superscript on $R$ represents the dimensions of the vector $\vec{x}$ .

$n$ indicates the number of rows.
$1$ indicates the number of columns.
$\times$ separates the number of rows from the number of columns. $⧫$

Remark. The subscript of a vector is used as an index variable. The first entry of a column vector is labeled $x_{1}$ , and we increase the index variable by one as we move downwards through the rows. The entry in the last row is called $x_{n}$ .
$⧫$