Vector Space

Table of contents
  1. Matrix as a Group
  2. Definition
  3. Vector Subspace
    1. Nullspace (Kernel)
  4. Dimension of Vector Space

Matrix as a Group

Read more about groups here.

  • $(\mathbb{R}^{m \times n}, +)$ is an Abelian group
  • $(\mathbb{R}^{n \times n}, \cdot)$ is a general linear group $GL(n, \mathbb{R})$, if the square matrix is invertible (regular, non-singular)
    • Closure, associativity, neutral element (identity matrix) are already satisfied
    • However, it is not commutative, so it is not an Abelian group

Definition

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

$$ \begin{align*} +: \mathcal{V} \times \mathcal{V} &\to \mathcal{V} \\ \cdot: \mathbb{R} \times \mathcal{V} &\to \mathcal{V} \end{align*} $$

where:

  • $(\mathcal{V}, +)$ is an Abelian group
  • Distributive
    • In short, stuff like $\lambda \cdot (\mathbf{x} + \mathbf{y})$ and $(\lambda + \psi) \cdot \mathbf{x}$
  • Associative (scalar multiplication only)
    • $(\lambda \psi) \cdot \mathbf{x} = \lambda \cdot (\psi \cdot \mathbf{x})$
  • Neutral element with respect to scalar multiplication
    • $1 \cdot \mathbf{x} = \mathbf{x}$

Then $\mathbf{x} \in V$ is called vector.

Vector Subspace

  • Subspace is contained in a vector space, but has closure within itself.
  • Also called linear subspace.

Let $V = (\mathcal{V}, +, \cdot)$ be a vector space and $\mathcal{U} \subseteq \mathcal{V}$ and $\mathcal{U} \neq \emptyset$.
Then $U = (\mathcal{U}, +, \cdot)$ is a vector subspace of $V$, or $U \subseteq V$.

As long as the following holds:

  • $\mathbf{0} \in \mathcal{U}$
  • Closure of $U$ with respect to both inner and outer operations

Just like subsets, the trivial subspace of any vector space $V$ is $\{\mathbf{0}\}$ and $V$ itself.

The intersection of any number of subspaces of a vector space $V$ is also a subspace of $V$.

Nullspace (Kernel)

The solution space of a homogeneous system of linear equation (also called nullspace or kernel) $A \mathbf{x} = \mathbf{0}$ is a vector subspace of $\mathbb{R}^n$.

However, the solution space of a inhomogeneous system of linear equation $A \mathbf{x} = \mathbf{b}$ where $\mathbf{b} \neq \mathbf{0}$ is not a vector subspace of $\mathbb{R}^n$.

Because the zero vector is not in the solution space of an inhomogeneous system.

Dimension of Vector Space

Basis and Rank