Standard basis ( $e_{1}, \dots, e_{n}$ )

The standard basis of the vector space

R^{n}

consists of the vectors

e_{i}

whose

i

th coordinate is one and whose other coordinates are zero. Every vector

x

has the unique expression

x = \sum_{i} x_{i} e_{i}

in this basis.

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- Standard basis vector Standard basis

Standard basis vector ( $e_{i}$ )

The standard basis vector

e_{i}

in

F^{n}

has coordinate one in position

i

and zero elsewhere. Every vector

v = (v_{1}, \dots, v_{n})

has the unique expansion

v = \sum_{i = 1}^{n} v_{i} e_{i}

.

For example,

e_{2} = (0, 1, 0)^{⊤}

in three dimensions. The outer product

e_{i} e_{j}^{⊤}

is the matrix unit with a single one in row

i

, column

j

, providing a standard basis for matrices as well. The word standard refers to the chosen coordinate representation; an abstract vector space does not come with a distinguished coordinate basis automatically.

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Standard basis by

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In linear algebra, the term "standard basis" typically refers to a set of basis vectors that provide a simple and intuitive way to understand vector spaces. The standard basis differs based on the context, usually depending on whether the vector space is defined over the real numbers \( \mathbb{R}^n \) or the complex numbers \( \mathbb{C}^n \).

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