Skip to main content

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 $x \in \mathbb{R}^n$

Comments

Popular posts from this blog

Mathematics for Machine Learning- CST 284 - KTU Minor Notes - Dr Binu V P

  Introduction About Me Syllabus Course Outcomes and Model Question Paper University Question Papers and Evaluation Scheme -Mathematics for Machine learning CST 284 KTU Overview of Machine Learning What is Machine Learning (video) Learn the Seven Steps in Machine Learning (video) Linear Algebra in Machine Learning Module I- Linear Algebra 1.Geometry of Linear Equations (video-Gilbert Strang) 2.Elimination with Matrices (video-Gilbert Strang) 3.Solving System of equations using Gauss Elimination Method 4.Row Echelon form and Reduced Row Echelon Form -Python Code 5.Solving system of equations Python code 6. Practice problems Gauss Elimination ( contact) 7.Finding Inverse using Gauss Jordan Elimination  (video) 8.Finding Inverse using Gauss Jordan Elimination-Python code Vectors in Machine Learning- Basics 9.Vector spaces and sub spaces 10.Linear Independence 11.Linear Independence, Basis and Dimension (video) 12.Generating set basis and span 13.Rank of a Matrix 14.Linear Mapping...

4.3 Sum Rule, Product Rule, and Bayes’ Theorem

 We think of probability theory as an extension to logical reasoning Probabilistic modeling  provides a principled foundation for designing machine learning methods. Once we have defined probability distributions corresponding to the uncertainties of the data and our problem, it turns out that there are only two fundamental rules, the sum rule and the product rule. Let $p(x,y)$ is the joint distribution of the two random variables $x, y$. The distributions $p(x)$ and $p(y)$ are the corresponding marginal distributions, and $p(y |x)$ is the conditional distribution of $y$ given $x$. Sum Rule The addition rule states the probability of two events is the sum of the probability that either will happen minus the probability that both will happen. The addition rule is: $P(A∪B)=P(A)+P(B)−P(A∩B)$ Suppose $A$ and $B$ are disjoint, their intersection is empty. Then the probability of their intersection is zero. In symbols:  $P(A∩B)=0$  The addition law then simplifies to: $P(...

5.1 Optimization using Gradient Descent

Since machine learning algorithms are implemented on a computer, the mathematical formulations are expressed as numerical optimization methods.Training a machine learning model often boils down to finding a good set of parameters. The notion of “good” is determined by the objective function or the probabilistic model. Given an objective function, finding the best value is done using optimization algorithms. There are two main branches of continuous optimization constrained and unconstrained. By convention, most objective functions in machine learning are intended to be minimized, that is, the best value is the minimum value. Intuitively finding the best value is like finding the valleys of the objective function, and the gradients point us uphill. The idea is to move downhill (opposite to the gradient) and hope to find the deepest point. For unconstrained optimization, this is the only concept we need,but there are several design choices. For constrained optimization, we need to intr...