Definition:Vector (Linear Algebra)

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This page is about vector in the context of vector spaces. For other uses, see Definition:Vector.

Definition

Let $V$ be a vector space.


Any element $v$ of $V$ is called a vector.


Vector Notation

Several conventions are found in the literature for annotating a general vector in a style that distinguishes it from a scalar, as follows.

Let $\set {x_1, x_2, \ldots, x_n}$ be a collection of scalars which form the components of an $n$-dimensional vector.

The vector $\tuple {x_1, x_2, \ldots, x_n}$ can be annotated as:

\(\displaystyle \bsx\) \(=\) \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\)
\(\displaystyle \vec x\) \(=\) \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\)
\(\displaystyle \hat x\) \(=\) \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\)
\(\displaystyle \underline x\) \(=\) \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\)
\(\displaystyle \tilde x\) \(=\) \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\)


To emphasize the arrow interpretation of a vector, we can write:

$\bsv = \sqbrk {x_1, x_2, \ldots, x_n}$

or:

$\bsv = \sequence {x_1, x_2, \ldots, x_n}$


In printed material the boldface $\bsx$ is common. This is the style encouraged and endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$.

However, for handwritten material (where boldface is difficult to render) it is usual to use the underline version $\underline x$.

Also found in handwritten work are the tilde version $\tilde x$ and arrow version $\vec x$, but as these are more intricate than the simple underline (and therefore more time-consuming and tedious to write), they will only usually be found in fair copy.

It is also noted that the tilde over $\tilde x$ does not render well in MathJax under all browsers, and differs little visually from an overline.

The hat version $\hat x$ usually has a more specialized meaning, namely to symbolize a unit vector.


In computer-rendered materials, the arrow version $\vec x$ is popular, as it is descriptive and relatively unambiguous, and in $\LaTeX$ it is straightforward.

However, it does not render well in all browsers, and is therefore (reluctantly) not recommended for use on this website.

Because of this method of rendition, some sources refer to vectors as arrows.


Also see


Sources