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Linear Functionals - Linear Forms - 1/One-Forms - Covectors

Linear Functionals - Linear Forms - 1/One-Forms - Covectors

Jan 12, 2024

Linear Functionals - Linear Forms - 1/One-Forms - Covectors (𝐿: 𝑉 → 𝐹)

Linear Functional - Examples

General Linear Functional ExamplesDescription
Linear Functionals in ℝ𝑛
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    Given a vector space 𝑉 over a field 𝐹, let 𝑣 be a column vector in 𝑉:

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    For each row vector 𝑎 = [𝑎1, 𝑎2, ..., 𝑎𝑛] there is a linear functional 𝑓𝑎 defined by:

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    and each linear functional can be expressed in this form.

    This can be interpreted as either the matrix product or the dot product of the row vector 𝑎 and the column vector 𝑣:

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Definite Integrals

Specific Linear Functionals in ℝ𝑛Description
zero function
  • one-form 𝑎 = [0, 0, …, 0] = zero row-vector, maps every column-vector to scalar 0
  • is the ONLY linear function that is non-surjective, every other linear functional is surjective
indexing into a vector
  • one-form 𝑎 = [0,1,0]. That is, the second element of [𝑥,𝑦,𝑧] is [0,1,0]·[𝑥,𝑦,𝑧] = 𝑦
mean
  • one-form 𝑎 = [1/𝑛, 1/𝑛, …, 1/𝑛] where 𝑛 is the number of elements. That is, 𝑚𝑒𝑎𝑛⁡(𝑣) = [1/𝑛, 1/𝑛, …, 1/𝑛]·𝑣

Linear Functional - Non-Examples

  • A function 𝑓 having the equation of a line 𝑓(𝑥) = 𝑎 + 𝑟𝑥 with 𝑎 ≠ 0. For example, 𝑓(𝑥) = 1 + 2𝑥) is not a linear functional on ℝ, since it is not linear. It is however affine-linear

Linear Functional - Visualizations

  • level sets can be used to visualize linear functionals

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