Examples of vector spaces: Difference between revisions

Revision as of 00:13, 8 June 2018

Revision as of 16:59, 18 October 2018

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:<math>0 = (0, 0, \ldots, 0) </math>

:<math>-x = (-x_1, -x_2, \ldots, -x_n) </math>

The most common cases are whereCommonly, '''F''' is the field of [[real number]]s, givingin thewhich case we obtain [[real coordinate space]] '''R'''^''n'',. or theThe field of [[complex number]]s giving thegives [[complex coordinate space]] '''C'''^''n''. The ''a + bi'' form of a complex number shows that '''C''' itself is a two-dimensional real vector space with coordinates (''a'',''b''). Similarly, the [[quaternion]]s and the [[octonion]]s are respectively four- and eight-dimensional real vector spaces, and '''C'''^''n'' is a ''2n''-dimensional real vector space.

The [[quaternion]]s and the [[octonion]]s are respectively four- and eight-dimensional vector spaces over the reals.

The vector space '''R'''^''n'' comes with a [[standard basis]]: