Vector Notation

# General Vector Notation

There are two common ways to denote a vector. The first way is with the use of an arrow such as $\vec{u}$. However, it is also common to use boldface such as $\mathbf{u}$. In the linear algebra section of MathOnline the former notation will be used much more frequently.

# Component Form of a Vector

The component form of a vector in Euclidean n-space is commonly denoted as $\vec{u} = (u_{1}, u_{2}, ... , u_{n})$, $\vec{u} \in \mathbb{R} ^n$ where $u_1, u_2, ..., u_n$ are said to be the components of $\vec{u}$, while the notation "$\vec{u} \in \mathbb{R} ^n$" says that $\vec{u}$ exists within Euclidean n-space.

# Matrix Form of a Vector

A vector can also be represented in terms of a $m \times 1$ matrix known as a column matrix or column vector. Alternatively, we can represent a vector is a $1 \times n$ matrix known as a row matrix or row vector. For example, the column-matrix form of a vector would look as follows:

(1)
\begin{align} \vec{u} = \begin{bmatrix} u_{1}\\ u_{2}\\ \vdots \\ u_{n}\end{bmatrix} , \end{align}

While the row-matrix form of a vector would look like this:

(2)
\begin{align} \vec{u} = \begin{bmatrix} u_{1} & u_{2}& ... & u_{n} \end{bmatrix} \end{align}