Physics Lournal

Numerical Field

A field, is a set $F$ of numbers, which is equipped with the property that if $a,b \in F$ , then:

$a + b, a -b, a(b)\; \text{and}\; a/b\; \text{are also in}\; F$ , with the caveat that $b \geq 1$ for division.

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.

$\N\; and\; \R$ are sets of Natural and Real numbers, but not all sets of numbers are fields of numbers: $\N$ is not a field because it contains some $a,b$ , such as $3, 5,$ that does not satisfy the definition, as $3 - 5 \notin \N$ .

$N$ refers to the set of Natural (counting) numbers, which does not contain any negative numbers, hence why it is a set of numbers that cannot satisfy the definition of a field.

$\R$ is a field because it contains negative and positive numbers, which means no elements of it to which the required operations are applied can yield a number of a signed-ness that does not exist within it.

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 $\R^{2}$ .

Referenced in

Chapter 1: Vector Spaces

Throughout this book, we will use $\mathbb{F}$ , as a stand in for $\mathbb{R}, \text{and}\; \mathbb{C}$ , as both of these are examples of fields.

Vector Space

Understanding vector spaces first requires understanding of the concept of a Field, of numbers.

Quantum Information Theory

An inner product on a vector space (defined over the field of complex numbers), is a function that relates a pair of vectors $|u\rangle, |v\rangle \in V$ with a complex number $\langle u|v\rangle$ , such that the following axioms are satisfied:

Numerical Field

Numerical Field

A field, is a set FFF of numbers, which is equipped with the property that if a,b∈Fa,b \in Fa,b∈F, then:

a+b,a−b,a(b) and a/b are also in Fa + b, a -b, a(b)\; \text{and}\; a/b\; \text{are also in}\; Fa+b,a−b,a(b)anda/bare also inF, with the caveat that b≥1b \geq 1b≥1 for division.

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.

N and R\N\; and\; \RNandR are sets of Natural and Real numbers, but not all sets of numbers are fields of numbers: N\NN is not a field because it contains some a,ba,ba,b, such as 3,5,3, 5,3,5, that does not satisfy the definition, as 3−5∉N3 - 5 \notin \N3−5∈/N.

NNN refers to the set of Natural (counting) numbers, which does not contain any negative numbers, hence why it is a set of numbers that cannot satisfy the definition of a field.

R\RR is a field because it contains negative and positive numbers, which means no elements of it to which the required operations are applied can yield a number of a signed-ness that does not exist within it.

To clarify this, we will do an example, demonstrating the set R2,\R^{2},R2, over a field R\RR is a vector space, meaning:

To clarify this, we will do an example, demonstrating the set R2,\R^{2},R2, over a field R\RR is a vector space, meaning:

(x1 y1)+(x2 y2)=(x1+y2 x2+y2)\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}(x1​ y1​​)+(x2​ y2​​)=(x1​+y2​ x2​+y2​​), is also contained in R2\R^{2}R2.

(x1 y1)+(x2 y2)=(x1+y2 x2+y2)\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}(x1​ y1​​)+(x2​ y2​​)=(x1​+y2​ x2​+y2​​), is also contained in R2\R^{2}R2.

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 R2\R^{2}R2.

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 R2\R^{2}R2.

Referenced in

Chapter 1: Vector Spaces

Throughout this book, we will use F\mathbb{F}F, as a stand in for R,and C\mathbb{R}, \text{and}\; \mathbb{C}R,andC, as both of these are examples of fields.

Vector Space

Understanding vector spaces first requires understanding of the concept of a Field, of numbers.

Quantum Information Theory

An inner product on a vector space (defined over the field of complex numbers), is a function that relates a pair of vectors ∣u⟩,∣v⟩∈V|u\rangle, |v\rangle \in V∣u⟩,∣v⟩∈V with a complex number ⟨u∣v⟩\langle u|v\rangle⟨u∣v⟩, such that the following axioms are satisfied:

A field, is a set $F$ of numbers, which is equipped with the property that if $a,b \in F$ , then:

$a + b, a -b, a(b)\; \text{and}\; a/b\; \text{are also in}\; F$ , with the caveat that $b \geq 1$ for division.

$\N\; and\; \R$ are sets of Natural and Real numbers, but not all sets of numbers are fields of numbers: $\N$ is not a field because it contains some $a,b$ , such as $3, 5,$ that does not satisfy the definition, as $3 - 5 \notin \N$ .

$N$ refers to the set of Natural (counting) numbers, which does not contain any negative numbers, hence why it is a set of numbers that cannot satisfy the definition of a field.

$\R$ is a field because it contains negative and positive numbers, which means no elements of it to which the required operations are applied can yield a number of a signed-ness that does not exist within it.

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

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 $\R^{2}$ .

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

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 $\R^{2}$ .

Throughout this book, we will use $\mathbb{F}$ , as a stand in for $\mathbb{R}, \text{and}\; \mathbb{C}$ , as both of these are examples of fields.

An inner product on a vector space (defined over the field of complex numbers), is a function that relates a pair of vectors $|u\rangle, |v\rangle \in V$ with a complex number $\langle u|v\rangle$ , such that the following axioms are satisfied: