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Part of Qiskit: Intro to Linear Algebra

To clarify this, we will do an example, demonstrating the set $\R^{2},$ over a field $\R$ is a vector space, meaning:

$\footnotesize \begin{pmatrix} x_{1}\ y_{1} \end{pmatrix} + \begin{pmatrix} x_{2}\ y_{2} \end{pmatrix} = \begin{pmatrix} x_{1} + y_{2} \ x_{2} + y_{2} \end{pmatrix}$ , is also contained in $\R^{2}$ .

This is true because the sum of two real numbers is also a real number, meaning both new vector components are real numbers, and so they must reside in $\R^{2}$ .

Sidenote: A field in $\R^{2}$ is written as $\R$ , because it's a plane.

Referenced in

Numerical Field

Fields can be thought of as sets for these purposes, but they are technically defined as Commutative Rings, and their elements are referred to as scalars.

Qiskit: Intro to Linear Algebra