Physics Lournal

Vector Space

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

From this concept, the concept of a vector space can be defined:

An addition operation, referred to as vector addition, which takes two vectors $u ,v\in V$ , and produces some third vector $w$ , which is equal to $u + v$ , and exists in $V, (w \in V)$ .

Worked Example: Associativity.

$\begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix} + \begin{pmatrix} y_1 \ y_2 \ \vdots \ y_n \end{pmatrix} + \begin{pmatrix} z_1 \ z_2 \ \vdots \ z_n \end{pmatrix} = \begin{pmatrix} x_1 + y_1 \ x_2 + y_2 \ \vdots \ x_n + y_n\ \end{pmatrix} + \begin{pmatrix} z_1 \ z_2 \ \vdots \ z_n \end{pmatrix}$

$\begin{pmatrix} (x_1 + y_1) + z_1 \ (x_2 + y_2 ) + z_2\ \vdots \ (x_n + y_n) + z_n \ \end{pmatrix}$

$\begin{pmatrix} x_1 + (y_1 + z_1) \ x_2 + (y_2 + z_2)\ \vdots \ x_n + (y_n + z_n) \ \end{pmatrix}$

A scalar multiplication operation that takes some scalar $c$ , which is an element of $F$ , ( $c \in F$ ), and a vector $v \in V$ , and produces a new vector $w$ , which is equal to $c(v)$ , and exists in $\small V, (w \in V)$ .

$c\begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix} = \begin{pmatrix} c(x_1) \ c(x_2) \ \vdots \ c(x_n) \end{pmatrix} = \begin{pmatrix} cx_1 \ cx_2 \ \vdots \cx_n \end{pmatrix} = w$ .

The above operations, however, are not all that is required to define a vector space, as vector spaces must also satisfy a set of Axioms:

Vector addition must display associativity, meaning for any $u, v \in V$ , the addition of them to yield $w$ , must not yield a different vector, if the two addend vectors are swapped:

$u + v = w, v + u = w$ , for all $u, v, w \in V$ .

Worked Example: Associativity.

$\begin{pmatrix} (x_1 + y_1) + z_1 \ (x_2 + y_2 ) + z_2\ \vdots \ (x_n + y_n) + z_n \ \end{pmatrix}$

$\begin{pmatrix} x_1 + (y_1 + z_1) \ x_2 + (y_2 + z_2)\ \vdots \ x_n + (y_n + z_n) \ \end{pmatrix}$

There must exist a zero vector, written $0$ , meaning there is some vector in $V$ , such that $0 + v = v$ for all $v \in V$ .

$0 + \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n\end{pmatrix} = \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n\end{pmatrix}$

For every $u \in V$ , there must be a vector written $-u$ , such that $u + (-u) = 0$ .

$\begin{pmatrix} y_1 \ y_2 \ \vdots \ y_n \end{pmatrix} + \begin{pmatrix} (-y_1) \ (-y_2) \ \vdots \ (-y_n) \end{pmatrix} = 0$

Associativity: For every $a, b \in F$ and any vector $u \in V$ , $(ab)u = a(bu)$ .

$a(b) \begin{pmatrix} y_1 \ y_2 \ \vdots \ y_n \end{pmatrix} = a \begin{pmatrix} b(y_1) \ b(y_2) \ \vdots \ b(y_n) \end{pmatrix}$

Distributivity: For every $a, b \in F$ and any vectors $u, v \in V$ :

$(a + b)u = au + bu, \text{and}\; a(u + v) = au + av$ .

$(a +b) \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n\end{pmatrix} = a\begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n\end{pmatrix} + b\begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n\end{pmatrix}$

$a\begin{pmatrix} x_1 + y_1 \ x_2 + y_2 \ \vdots \ x_n + y_n\ \end{pmatrix} = a\begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n\end{pmatrix} + a\begin{pmatrix} y_1 \ y_2 \ \vdots \ y_n \end{pmatrix}$

Unitarity: $1(u) = u, \forall u \in V$ .

$1\begin{pmatrix} x_1 \ x_2\ \vdots \ x_n \end{pmatrix} = \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix}$

$\forall$ is read as "For all".

Referenced in

Quantum Information Theory

More specifically, a complex vector space, is a set $V$ , such that any two vectors $a,b$ , (which are elements of the space: ( $\alpha, \beta \in V$ )), can be combined to form the following linear combination:

The Principle Of Superposition

Dimensions

The dimension of some mathematical space (such as Vector Space), is informally defined as the minimal number of coordinates to describe some point that exists within the space.

Qiskit: Intro to Linear Algebra

A vector $| v \rangle$ , is defined as elements of a set known as a Vector Space.

Hilbert Space

Hilbert Space is an infinite dimensional Vector Space, generalized over Euclidean Space.

Chapter 1: Vector Spaces

Linear algebra is the study of Linear Maps on finite-dimensional vector spaces.

Birds Eye View of Quantum Mechanics

In more technical language: the eigenstates of some operator form the basis vectors of the Vector Space that contains our state vectors.

Mathematical Interlude - Vector Spaces

Within classical systems, the logic is boolean, and the state space is a set, but with quantum systems, these states are represented by a vector space, particularly a special type of Vector Space, known as a Hilbert Space.

Linear Map

In Mathematics, a linear map, is a mapping $V \rightarrow W$ , between two vector spaces, that preserves the results of the operations of vector addition, and scalar multiplication.

Vector Space

Vector Space

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

From this concept, the concept of a vector space can be defined:

A vector space consists of a set of elements VVV, the elements of which are referred to as vectors, which is related to some field FFF, and equipped with two operations:

An addition operation, referred to as vector addition, which takes two vectors u,v∈Vu ,v\in Vu,v∈V, and produces some third vector www, which is equal to u+vu + vu+v, and exists in V,(w∈V)V, (w \in V)V,(w∈V).

Worked Example: Associativity.

A scalar multiplication operation that takes some scalar ccc, which is an element of FFF, (c∈Fc \in Fc∈F), and a vector v∈Vv \in Vv∈V, and produces a new vector www, which is equal to c(v)c(v)c(v), and exists in V,(w∈V)\small V, (w \in V)V,(w∈V).

The above operations, however, are not all that is required to define a vector space, as vector spaces must also satisfy a set of Axioms:

Vector addition must display associativity, meaning for any u,v∈Vu, v \in Vu,v∈V, the addition of them to yield www, must not yield a different vector, if the two addend vectors are swapped:

u+v=w,v+u=wu + v = w, v + u = wu+v=w,v+u=w, for all u,v,w∈Vu, v, w \in Vu,v,w∈V.

Worked Example: Associativity.

There must exist a zero vector, written 000, meaning there is some vector in VVV, such that 0+v=v0 + v = v0+v=v for all v∈Vv \in Vv∈V.

For every u∈Vu \in Vu∈V, there must be a vector written −u-u−u, such that u+(−u)=0u + (-u) = 0u+(−u)=0.

Associativity: For every a,b∈Fa, b \in Fa,b∈F and any vector u∈Vu \in Vu∈V, (ab)u=a(bu)(ab)u = a(bu)(ab)u=a(bu).

Distributivity: For every a,b∈Fa, b \in Fa,b∈F and any vectors u,v∈Vu, v \in Vu,v∈V:

(a+b)u=au+bu,and a(u+v)=au+av(a + b)u = au + bu, \text{and}\; a(u + v) = au + av(a+b)u=au+bu,anda(u+v)=au+av.

Unitarity: 1(u)=u,∀u∈V1(u) = u, \forall u \in V1(u)=u,∀u∈V.

1(x1 x2 ⋮ xn)=(x1 x2 ⋮ xn)1\begin{pmatrix} x_1 \ x_2\ \vdots \ x_n \end{pmatrix} = \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix}1(x1​ x2​ ⋮ xn​​)=(x1​ x2​ ⋮ xn​​)

∀\forall∀ is read as "For all".

Referenced in

Quantum Information Theory

More specifically, a complex vector space, is a set VVV, such that any two vectors a,ba,ba,b, (which are elements of the space: (α,β∈V\alpha, \beta \in Vα,β∈V)), can be combined to form the following linear combination:

The Principle Of Superposition

Dimensions

The dimension of some mathematical space (such as Vector Space), is informally defined as the minimal number of coordinates to describe some point that exists within the space.

Qiskit: Intro to Linear Algebra

A vector ∣v⟩| v \rangle∣v⟩, is defined as elements of a set known as a Vector Space.

Hilbert Space

Hilbert Space is an infinite dimensional Vector Space, generalized over Euclidean Space.

Chapter 1: Vector Spaces

Linear algebra is the study of Linear Maps on finite-dimensional vector spaces.

Birds Eye View of Quantum Mechanics

In more technical language: the eigenstates of some operator form the basis vectors of the Vector Space that contains our state vectors.

Mathematical Interlude - Vector Spaces

Within classical systems, the logic is boolean, and the state space is a set, but with quantum systems, these states are represented by a vector space, particularly a special type of Vector Space, known as a Hilbert Space.

Linear Map

In Mathematics, a linear map, is a mapping V→WV \rightarrow WV→W, between two vector spaces, that preserves the results of the operations of vector addition, and scalar multiplication.

A vector space consists of a set of elements $V$ , the elements of which are referred to as vectors, which is related to some field $F$ , and equipped with two operations:

An addition operation, referred to as vector addition, which takes two vectors $u ,v\in V$ , and produces some third vector $w$ , which is equal to $u + v$ , and exists in $V, (w \in V)$ .

A scalar multiplication operation that takes some scalar $c$ , which is an element of $F$ , ( $c \in F$ ), and a vector $v \in V$ , and produces a new vector $w$ , which is equal to $c(v)$ , and exists in $\small V, (w \in V)$ .

Vector addition must display associativity, meaning for any $u, v \in V$ , the addition of them to yield $w$ , must not yield a different vector, if the two addend vectors are swapped:

$u + v = w, v + u = w$ , for all $u, v, w \in V$ .

There must exist a zero vector, written $0$ , meaning there is some vector in $V$ , such that $0 + v = v$ for all $v \in V$ .

For every $u \in V$ , there must be a vector written $-u$ , such that $u + (-u) = 0$ .

Associativity: For every $a, b \in F$ and any vector $u \in V$ , $(ab)u = a(bu)$ .

Distributivity: For every $a, b \in F$ and any vectors $u, v \in V$ :

$(a + b)u = au + bu, \text{and}\; a(u + v) = au + av$ .

Unitarity: $1(u) = u, \forall u \in V$ .

$1\begin{pmatrix} x_1 \ x_2\ \vdots \ x_n \end{pmatrix} = \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix}$

$\forall$ is read as "For all".

More specifically, a complex vector space, is a set $V$ , such that any two vectors $a,b$ , (which are elements of the space: ( $\alpha, \beta \in V$ )), can be combined to form the following linear combination:

A vector $| v \rangle$ , is defined as elements of a set known as a Vector Space.

In Mathematics, a linear map, is a mapping $V \rightarrow W$ , between two vector spaces, that preserves the results of the operations of vector addition, and scalar multiplication.