# Definition:Vector/Linear Algebra

## 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

- 1961: I.M. Gel'fand:
*Lectures on Linear Algebra*(2nd ed.) ... (previous) ... (next): $\S 1$: $n$-Dimensional vector spaces - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**vector space** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem