Definition:Vector/Linear Algebra

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Definition

Let $V = \struct {G, +_G, \circ}_K$ be a vector space over $K$, where:

$\struct {G, +_G}$ is an abelian group
$\struct {K, +_K, \times_K}$ is the scalar field of $V$.


The elements of the abelian group $\struct {G, +_G}$ are called vectors.


Vector Notation

Several conventions are found in the literature for annotating a general vector quantity in a style that distinguishes it from a scalar quantity, 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:

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


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 $\bsx$ or $\mathbf x$ 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: $\overline x$.

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.


Also see


Sources

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