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Mathematics for Physics

240 Basis

A set of vectors that spans a vector space and represents every vector by unique scalar values that is used to write vectors in coordinates relative to that set.

definition [d] (Basis = Ordered Basis) An ordered set of linearly independent vectors that span a vector space \(V\).

where

  • \(V\) is a vector space.
  • the ordered set is the basis of \(V\).

Note:

  • two bases are considered different if one is a rearrangement of the other.
  • this ordered notion is sometimes called an ordered basis.

definition [d] (Basis) A set \(B\) of linearly independent vectors that spans all of a vector space \(V\).

where

  • \(V\) is a vector space.
  • \(B\) is the basis of \(V\).

Note:

  • every vector in \(V\) is a unique linear combination of the vectors in \(B\).
  • the coefficients in that expansion are the components relative to \(B\).

definition [d] (Basis) A set of vectors that both spans a vector space and is linearly independent: every vector is a linear combination of the basis vectors, and no basis vector is a linear combination of the others.

where

  • the spanning condition means every vector is a linear combination of the basis vectors.
  • the linear-independence condition means no basis vector is a linear combination of the remaining basis vectors.

Note:

  • this formulation is the geometric one used in spacetime treatments of tangent spaces.

240.1 Elementary Example

240.1.1 Simple

A basis is a linearly independent spanning set. The standard basis of \(\mathbb{R}^{2}\) has two vectors.

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

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

\[ v = v^{1} e_{1} + v^{2} e_{2} \]

where

  • \(B\) is the basis.
  • \(v^{1}, v^{2}\) are the unique coordinates of \(v\).

240.1.2 General

In \(\mathbb{R}^{3}\), three independent vectors form a basis and give unique coordinates.

\[ B = \{ e_{1},\ e_{2},\ e_{3} \} \]

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

\[ v = v^{1} e_{1} + v^{2} e_{2} + v^{3} e_{3} \]

where

  • every vector in \(\mathbb{R}^{3}\) has a unique expansion in \(B\).

240.2 References

  1. Dummit, D. S., & Foote, R. M. Abstract Algebra. Wiley, 2004. — ordered basis as ordered linearly independent spanning set.
  2. Hassani, S. Mathematical Physics, 2nd ed. Springer. — basis as linearly independent spanning set; unique components.
  3. Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — basis as spanning and linearly independent.