Functions of Vectors

A function is a map \(f:\mathcal{V}\to\mathcal{S}\).

Easy to imagine two useful categories of such functions: field elements or vectors. You can map vectors to arbitrary sets.

Functional

A map from a vector to a field element is defined as a functional.

Operator

A map from a vector to a vector is defined as a operator.

Linear Maps

A map from any vector space \(\mathcal{V}\) is linear if it distributes over the vector space operations:

  • \(f(\alpha|a\rangle + \beta|b\rangle) = \alpha f(|a\rangle) + \beta f(|b\rangle)\)

Linear Functionals - Dual Space

Definition

\(\mathcal{V}^*\) is the dual space of \(\mathcal{V}\), where elements of \(\mathcal{V}^*\) are all linear functionals \(f:\mathcal{V}\to\mathbb{F}\).

Notation: Define \(f(|a\rangle) \equiv \langle f|a\rangle\in\mathbb{F}\).

The Dual Space is a Vector Space.

Linear Operators

\(F(|a\rangle) = F|a\rangle=|Fa\rangle\)

Examples

  1. \(M: \mathbb{R}^n\to\mathbb{R}^m\). \(m\times n\) matrices.
  2. \(M: \mathcal{V}\to\mathcal{V}\), with \(\mathcal{V}\) being a funcation space. Derivatives, fourier transform, Laplace, integral.

Anti-Linear Maps

If \(\mathbb{F} = \mathbb{C}\) and \(f(\alpha|a\rangle + \beta |b\rangle) = \alpha^*f(|a\rangle) + \beta^* f(|b\rangle)\) then the operator is called anti-linear.

Author: Christian Cunningham

Created: 2023-06-25 Sun 02:25

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