Vector Space

Definition

$(V,F,+,\cdot )$ or $V$ is a vector space over $F$.

A Module whose set of scalar is a Field. $(V,+)$ is an Abelian Group scalar multiplication $\cdot$ is a Monoid action of $(F,\cdot )$ on $V$ the scalar multiplication is distributive with both vector and scalar addition

Vector

Where we call the element of vector space $v\in V$ a Vector

Dimension

Dimension (Vector Space)

Definition

$\dim (V)$ where $V$ is a Vector Space

The number of vectors in a basis for the Vector Space

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Conditions for Vector Space

Let $\mathbf{x},\mathbf{y}\in \mathbf{V}$

• $\mathbf{x}+\mathbf{y}\in \mathbf{V}$
• $k\mathbf{x}\in \mathbf{V}$
• $0\in \mathbf{V}$ where $\mathbf{o}$ is the origin.

$\Rightarrow {\sum }_{i=1}^n{k}_i\mathbf{x}\in \mathbf{V}$ (Linear Combination of Vectors)