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Mathematics for Physics

266 Standard Basis

A basis of unit vectors along the coordinate axes of an n-tuple space that is used to write vectors with the simplest coordinate values.

definition [d] (Standard Basis) The ordered list of vectors \(\hat{e}_{1},\ldots,\hat{e}_{n}\) in \(\mathbb{C}^{n}\), where \(\hat{e}_{j}\) is the \(n\)-tuple with a \(1\) in position \(j\) and zeros elsewhere. The same list is the standard basis of \(\mathbb{R}^{n}\).

where

  • \(n\) is a positive integer.
  • \(\hat{e}_{j}\) is the \(j\)-th standard basis vector.
  • \(\mathbb{C}^{n}\) is the space of complex \(n\)-tuples.
  • \(\mathbb{R}^{n}\) is the space of real \(n\)-tuples.

Note:

  • \(\hat{e}_{j}\) may equally be written \(e_{j}\).
  • the standard basis of \(\mathbb{C}^{n}\) is also a basis of \(\mathbb{R}^{n}\).

definition [d] (Standard Basis = Distinguished Basis) The distinguished basis of \(\mathbb{R}^{n}\) consisting of the column \(n\)-tuples

  • \((1,0,\ldots,0)^{\mathrm{T}},\; \ldots,\; (0,\ldots,0,1)^{\mathrm{T}}\) .

where

  • \(\mathbb{R}^{n}\) is the space of real \(n\)-tuples.
  • \((\,\cdot\,)^{\mathrm{T}}\) denotes the column vector obtained by transposition.

Note:

  • a general \(n\)-dimensional vector space need not come with any prescribed basis.
  • \(\mathbb{R}^{n}\) as the space of \(n\)-tuples is equipped with this distinguished basis.

266.1 Elementary Example

266.1.1 Simple

The standard basis of \(\mathbb{R}^{2}\) is the pair of unit axis vectors.

\[ E = \{ e_{1},\ e_{2} \} \]

\[ e_{1} = (1,0),\quad e_{2} = (0,1) \]

where

  • \(E\) is the standard basis.
  • each \(e_{i}\) points along a coordinate axis.

266.1.2 General

In \(\mathbb{R}^{n}\), the standard basis has \(n\) vectors with a single \(1\) in position \(i\).

\[ e_{1} = (1,0,0),\quad e_{2} = (0,1,0),\quad e_{3} = (0,0,1) \]

\[ \text{in }\mathbb{R}^{3} \]

where

  • \(e_{i}\) has a \(1\) in the \(i\)-th place and zeros elsewhere.

266.2 References

  1. Hassani, S. Mathematical Physics, 2nd ed. Springer. — standard basis of \(\mathbb{C}^{n}\) and of \(\mathbb{R}^{n}\).
  2. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — distinguished basis of \(\mathbb{R}^{n}\).