1.5 Vector Spaces and subspaces

Vector Spaces

A real valued vector space $\mathbb{V}= (\textit{V},+, \cdot)$ is a set $\textit{V}$ with two operations

$+: \textit{V} \times \textit{V} \implies {V}$

$\cdot : \mathbb{R} \times \textit{V} \implies {V}$

$+$ and $\cdot$ are standard vector addition and scalar multiplication where

1.$(\textit{V},+)$ is an Abelian group

2. Distributivity of scalar multiplication

 $\forall \lambda \in \mathbb{R} \quad and \quad x,y \in V: \lambda.(x+y)=\lambda.x + \lambda.y$

$\forall \lambda.\psi  \in \mathbb{R} \quad and \quad x,y \in V:( \lambda + \psi).(x+y)=\lambda.x + \psi.y$

3.Associativity of scalar multiplication

$\forall \lambda.\psi  \in \mathbb{R} \quad and \quad x \in V: \lambda  (\psi.x)=(\lambda.\psi).x$

4.Identity element with respect to multiplication

$\forall  x \in V: 1.x=x$

The elements $x \in V$ are called vectors.The identity element of $(V,+)$ is the zero vector $0=[0,\ldots,0]^T$.The operation $+$ is the vector addition.The elements $\lambda \in \mathbb{R}$ are called scalars and the operation $\cdot$ is a multiplication by scalar.

We write $x$ to denote a column vector and $x^T$ to represent a row vector which is the transpose of $x$.

$V=\mathbb{R}^n, n \in \mathbb{N}$ is a vector space with operation defined as follows

addition: $x+y=(x_1,x_2,\ldots,x_n)+(y_1,y_2,\ldots,y_n)=(x_1+y_1,x_2+y_2,\ldots,x_n+y_n)$ for all $x,y \in \mathbb{R}^n$

multiplication by scalar:$\lambda x =\lambda (x_1,\ldots,x_n)=(\lambda x_1,\ldots,\lambda x_n)$ for all $\lambda \in \mathbb{R},x \in \mathbb{R}^n$

Example: Set of all vectors $(a,b) \in R^2$ ie ; $(R^2,+,*)$ forms a vector space. In general $V=R^n$ forms a vector space.

$v=P$ forms a vector space in $R$, where $P$ is the set of all polynomials in variable $x$ and coefficient from $R$.

$v=P_n$ forms a vector space in $R$, where $P_n$ is the set of all polynomials in variable $x$ of degree at most $n$ and coefficient from $R$.

Note: The set of all polynomials of degree $n$ does not form a vector space because vector addition is not closed.

The vectors in $R^2$ with addition and multiplication defined as

$(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+2y_2)$

$k(x_1,y_1)=(kx_1,ky_1)$

dose not form a vector space because addition is not commutative and hence it does not form an abelian group.

Vector Subspaces

Intuitively, they are sets contained in the original vector space with the property that when we perform vector space operations on elements within this subspace, we will never leave it. In this sense, they are “closed”. Vector subspaces are a key idea in machine learning and we  use vector subspaces for dimensionality reduction.

Let $\mathbb{V}= (\textit{V},+, \cdot)$ be a vector space and $U \subseteq V, U \ne \phi$.Then $\mathbb{U}=(U,+,\cdot)$ is called vector subspace of $\mathbb{V}$ ( or linear subspace) if $\mathbb{U}$ is a vector subspace with the vector space operations $+$ and $\cdot$ restricted to $U \times U$ and $\mathbb{R} \times U$.We write $\mathbb{U} \subseteq \mathbb{V}$ to denote the subspace of $\mathbb{V}$.

Since $U$ is a subspace of $V$ it satisfies the Abelian group properties.To determine whether $(U,+,\cdot)$ is a subspace of $V$, we still need to show

1.$U \ne \phi$, in particular $0 \in U$

2.Closure with respect to $\cdot$ : $\forall \lambda \in \mathbb{R} , \forall x \in U: \lambda x \in U$

3.Closure with respect to +: $\forall x,y \in U: x+y \in U$

Theorem:

Let $S$ be a subset of $R^n$. Then $S$ is a subspace of $R^n$ if and only if the following conditions hold:

(a) $S$ is non-empty.

(b) For any $a, b \in R$ and any $\vec{u},\vec{v} \in S$,

$a \vec{u} + b \vec{v} \in S$

Examples:

• For every vector space $V$, the trivial subspaces are $V$ itself and ${0}$.
• Set of all $3 × 3$ skew symmetric or symmetric matrices is a subspace of vector space $V$ defines as:
• $V = \{M3×3| M$ is a $3 × 3$ matrices with real entries$\}$
• The set $S = \{(x_1, x_2, x_3) \in R^3| x_1 − x_2 + x_3 = 0\}$ is a subspace of $R^3$.
• The set $S = \{(x_1, x_2, x_3) \in R^3| x_1 = x_2 = x_3\}$ is a subspace of $R^3$.
• The set $S = \{(x_1, x_2, x_3) \in R^3| x_1 +x_2 + x_3=1\}$ is NOT a subspace of $R^3$.

Theorem: Let $V(F)$ is a vector space and let $S$ be a non-empty subset of $V$. Then $L(S)$ is a subspace of $V$.

The solution set of a homogenious system of linear equations $Ax=0$ with $n$ unknowns $x=[x_1,\ldots,x_n]^T$ is the subspace of $\mathbb{R}^n$.

The solution set of a inhomogenious system of linear equations $Ax=b,b \ne 0$ is not a subspace of $\mathbb{R}^n$ .

The intersection of arbitrarily many subspace is a subspace itself.

The union of subspaces is not a subspace.( if one contained in other they form a subspace)

every subspace $U \subseteq (\mathbb{R}^n,+, \cdot)$ is the solution space of a homogeneous system of equations $Ax=0$ for $x \in \mathbb{R}^n$