Vector Spaces - RPIQuantumComputing/QuantumCircuits GitHub Wiki
What is a Vector Space?
A vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on V such that five properties are satisfied. These five properties are as follows, where the term $F$ refers to a given field, which for our purposes will either refer to the set of real numbers $R$ or the set of complex numbers $C$.
commutativity
$$ u+v=v+u \qquad \forall u,v \in V $$
associativity
$$ (u+v)+w = u+(v+w) \land (ab)v=a(bv) \qquad \forall u,v,w \in V \land \forall a,b \in F $$
additive identity
$$ \exists 0 \in V :v+0=v \qquad \forall v \in V $$
additive inverse
$$ \forall v \in V \qquad \exists w \in V : v+w = 0 $$
multiplicative identity
$$ 1v = v \qquad \forall v \in V $$
distributive properties
$$ a(u+v) = au + av \land (a+b)v = av + bv \qquad \forall a,b \in F \land \forall u,v \in V $$
In the context of a vector space, addition and scalar multiplication can be defined like so...
Addition: addition on a set $V$ is a function that assigns an element $u+v \in V$ to each pair of elements $u, v\in V$.
Scalar Multiplication: scalar multiplication on a set $V$ is a function that assigns an element $\lambda v\in V$ to each $\lambda \in F$ and each $v \in V$.
Scalar multiplication is defined by $F$, so when describing vector spaces it is common practice to describe the vector space $V$ by saying $V$ is a vector space over $F$, rather than simply saying $V$ is a vector space. The two main kinds of vector spaces, are the real vector space and the complex vector space and they can appropriately be defined as such...
Real Vector Space: a vector space over $R$.
Complex Vector Space: a vector space over $C$.
Vector Space Notation
$F^n$ Notation
$F^n$ is the most common example of a vector space, of field $F$ and dimensionality $n$. It can be represented as an $nxn$ matrix, and can be described more precisely with the following definition.
$F^n$ is the set of all lists of length $n$ of elements of $F$:
$$ F^n = \{(x_1,...,x_n) : x_j \in F \; | \; j = 1,...,n\} $$
For $(x_1,...,x_n) \in F^n$ and $j \in {1,...,n}$, we say that $x_j$ is the %j^{th}$ coordinate of $(x_1,...,x_n)$.
$F^S$ Notation
$F^S$ is another example of a vector space, which can be described using the following definition.
If $S$ is a set, then $F^S$ denotes the set of function form $S$ to $F$
For $f,g \in F^S$, the sum $f+g \in F^S$ is the function defined by
$$ (f+g)(x)=f(x)+g(x) \qquad \forall x \in S $$
- For $\lambda \in F$ and $f \in F^S$, the product $\lambda f \in F^S$ is he function defined by
$$ (\lambda f)(x) = \lambda f(x) \qquad \forall x\in S $$
Additive Inverse Notation
Let $v, w \in V$. Then
- $-v$ denotes the additive inverse of $v$
- $w-v$ is defined to be $w + (-v)$
Vector Subspaces
Addition of Subsets
A subset $U$ of $V$ is referred to as a subspace of $V$ if $U$ is also a vector space (using the same addition and scalar multiplication as on $V$) In more precise terms a subset $U$ of $V$ is a subspace of $V$ if and only if $U$ satisfies the following three conditions:
additive identity:
$$ 0 \in U $$
closed under addition:
$$ u,w \in U \implies u+w \in U $$
closed under scalar multiplication:
$$ a \in F \land u \in U \implies au \in U $$
Sums of Vector Subspaces
Given that vector subspaces are subsets of a vector space which themselves are vector spaces, to understand the definition how a sum of vector spaces is defined, it is important to understandhow the sum of subsets is defined.
Suppose $U_1,...,U_m$ are subsets of $V$. The sum of $U_1,...,U_m$, denoted $U_1+ \cdots +U_m$, is the set of all possible sums of elements of $U_1,...,U_m$. To be more precise,
$$ U_1 + \cdots + U_m = \{u_1 + \cdots + u_m : u_1 \in U_1,...,u_m \in U_m \}. $$
Of some important note is that the sum of subspaces $U_1,...,U_m$ of $V$ represents the smallest subspace of $V$ containing $U_1,...,U_m$.
When it comes to the sums of subspaces there is a special type of sum known as the direct sum. The direct sum of subspaces can be defined like so...
Suppose $U_1,...,U_m$ are subspaces of $V$.
The sum $U_1 + \cdots + U_m$ is called a direct sum if each element of $U_1 + \cdots + U_m$ can be written in only one wway as a sum $u_1 + \cdots + u_m$, where $u_j$ is in $U_j$.
If $U_1 + \cdots + U_m$ is a direct sum, then $U_1 \oplus U_m$ denotes $U_1 + \cdots + U_m$, with the $\oplus$ notation serving as an indication that this is a direct sum.
By definition, supposing $U1, ..., U_m$ are subspaces of V, then $U_1 + \cdots + U_m$ will be a direct sum if and only if the only way to write $0$ as a sum, $u_1+...+u_m$ where each $u_j$ is in the subpsace $U_j$, is by taking each $u_j$ equal to $0$. Additionally, supposing $U$ and $W$ are subspaces of $V$, then $U+W$ is a direct sum if and only if the intersection of $U$ and $W$, $U \cap W$, must be $\{0\}$.
Finite-Dimensional Vector Spaces
Linear Combinations
Linear Combination
A linear combination of a list $v_1,...,v_m$ of vectors in V is a vector of the form...
$$ a_1 v_1 + \cdots + a_m v_m $$
where $a_1,...,a_m \in F$
Span
The set of a linear combinations of a list of vectors $v_1,...,v_m$ in $V$ is called the span of $v_1,...,v_m$ in $V$ is called the span of $v_1,...,v_m$, denoted $span(v_1,...,v_m)$. In other words,
$$ span(v_1,...,v_m)=\{a_1 v_1 + \cdots + a_m v_m : a_1, ... ,a_m \in F \} $$
the span of the empty list of vectors is $\{ 0 \}$
The span of a list of vectors in $V$ is the smallest subspace of $V$ containing all the vectors in the list. If a span, $span(v_1,...,v_m)$, equals the vector space $V$, it is said that the list of vectors $v_1,...,v_m$ spans $V$.
A vector space is described as finite-dimensional if some list of vectors in it spans the vector space. A vector space is then referred to as infinite-dimensional if is not finite dimensional, and thus there exists no set of vectors in the vector space that spans the vector space.
Linear Independence and Dependence
Linear Independence
A list $v_1,...,v_m$ of vectors in $V$ is called linearly independent if the only choice of $a_1,...,a_m \in F$ that makes $a_1 v_1 + \cdots + a_m v_m$ equal $ is $a_1 = \cdots = a_m = 0$
The empty list $()$ is also linearly independent
Linear Dependence
A list of vectors in $V$ is described as linearly dependent if it is not linearly independent
This is to say that a list of vectors $v_1,...,v_m$ of vectors in $V$ is linearly dependent if there exists$ a_1,...,a_m \in F$, not all $, such that $a_1 v_1 + \cdots + a_m v_m = 0$
In a finite-dimension vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list of vectors.
Every subspace of a finite-dimensional vector space is finite-dimensional.
Dimension
To being to understand the definition of the dimension of a vector space, it is necessary to first understand what the basis of a vector space is. The basis of a vector space can be defined as follows...
A basis of $V$ is a list of vectors in $V$ that is linearly independent and spans $V$. This means A list $v_1,...,v_n$ of vectors in $V$ is a basis of $V$ if and only if every $v \in V$ can be written in the form $$ v = a_1 v_1 + \cdots + a_n v_n $$
where $a_1,...,a_n \in F$
Now that the definition of a basis has been established, there are several properties of bases which should be established. For one, every spanning list in a vector space can be reduced to a basis of the vector space. Additionally, every finite-dimensional vector space has a basis, and every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vectors space. Finally it should be noted that any two bases of a finite-dimensional vector space have the same length.
So now there is the main question, what is the dimension of a vector space. The dimension of a vector space and its notation can be defined like so...
The dimension of a finite-dimensional vector space is the length of any basis of the vector space
the dimension of $V$ (if $V$ is finite-dimensional) is denoted by $dim \; V$
Among the properties of vector space's dimension one is that if $V$ is finite-dimensional and $U$ is a subspace of $V$, then $dim \; U \leq dim \; V$. Additionally, supposing $V$ is finite-dimensional, every spanning list of vectors in $V$ with length $dim \; V$ will be a basis of $V$.
When it comes to comes to the sum of two subspaces, the dimension of the result can be defined as follows...
If $U_1$ and $U_2$ are subspaces of a finite-dimensional vector space, then
$$ dim(U_1+U_2)=dim \; U_1 + dim \; U_2 - dim(U_1 \cap U_2) $$
Works Cited
Special thanks to Sheldon Axler's "Linear Algebra Done Right". This page is essentially a summation of his his section on Vector Spaces and Subspaces, incorporating mainly his definitions of concepts and notation and bare minimum of information necessary to understand them.