Vector Operations

Vectors and Their Operations. A matrix with only one column is called a column vector, or simply a vector. For example, $\mathbf{u} = \left[\begin{array}{c} 1 \\ -2 \\ -5 \end{array}\right]$ and $\mathbf{v} = \left[\begin{array}{c} 2 \\ 5 \\ 6 \end{array}\right]$ are vectors. Then we can define the addition “ $\mathbf{u} + \mathbf{v}$ ” of two vectors, and the multiplications “ $c \mathbf{u}$ ” of a vector with a real number

, in which

is called a scalar. In the above example,

$\displaystyle \mathbf{u} + \mathbf{v} = \left[\begin{array}{c} 1 + 2 \\ -2... ...d{array}\right] = \left[\begin{array}{c} 3 \\ 3 \\ 1 \end{array}\right]$ and $\displaystyle \quad c \mathbf{u} = \left[\begin{array}{c} c \times (1) \\ c \times (-2) \\ c \times (-5) \end{array}\right]$

EXAMPLE 1. (This demonstration requires arrow.m to draw vectors as in the plot below.) Given $\mathbf{u} = \left[\begin{array}{c} 1 \\ 1 \end{array}\right]$ and $\mathbf{v} = \left[\begin{array}{c} -1 \\ 2 \end{array}\right]$ , find $2 \mathbf{u} + \mathbf{v}$ and $\mathbf{u} -2 \mathbf{v}$ .

n-Dimensional Vectors. The following vectors $\mathbf{a}$ and $\mathbf{b}$ are called a 2-dimensional and 3-dimensional vector, respectively.

$\displaystyle \mathbf{a} = \left[\begin{array}{c} 3 \\ -1 \end{array}\right]$ and $\displaystyle \quad \mathbf{b} = \left[\begin{array}{c} 2 \\ 3 \\ 4 \end... ...bf{u} = \left[\begin{array}{c} a_1 \\ \vdots \\ a_n \end{array}\right] \:$ in general. $\displaystyle$

A vector $\mathbf{u}$ with

real numbers is called an n-dimensional vector. We often call it simply “a vector in $\mathbb{R}^n$ .”

Matlab/Octave. In Matlab/Octave we can set a new vector in either of the following forms:

> u = [1; -2; -5]

> u = [1 -2 -5]'

If you type only “u = [1 -2 -5]”, it gives a “row” instead of a “column.” Then the apostrophe (') must be attached, which does a transposition. You can add a vector and multiply it by a scalar as follows:

> u + v

> 5 * u