Row Vector

Definition

For an $m\times n$ matrix

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right]$$

the vectors

$$\begin{array}{rl}{\mathbf{r}}_1& =[a_{11}{a}_{12}\cdots a_{1n}]\\ {\mathbf{r}}_2& =[a_{21}{a}_{22}\cdots a_{2n}]\\ ⋮& \\ {\mathbf{r}}_m& =[a_{m1}{a}_{m2}\cdots a_{mn}]\end{array}$$

in $R^n$ that are formed from the rows of $A$ are called the row vectors of $A$, and the vectors

$${\mathbf{c}}_1=\left[\begin{array}{c}{a}_{11}\\ a_{21}\\ ⋮\\ a_{m1}\end{array}\right],{\mathbf{c}}_2=\left[\begin{array}{c}{a}_{12}\\ a_{22}\\ ⋮\\ a_{m2}\end{array}\right],\cdots ,{\mathbf{c}}_n=\left[\begin{array}{c}{a}_{1n}\\ a_{2n}\\ ⋮\\ a_{mn}\end{array}\right]$$

in $R^m$ formed from the columns of $A$ are called the column vectors of $A$.