Subspace of Vector Space

21-06-2024 19:01

Status :

Tags : linear algebra Linear Equations Vector Spaces

Prerequisites

- Vector Spaces

- Sets and Relations

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Subspace of Vector Space

1.3.2 Subspace of Vector Space: If $(V)$ is a vector space and $(W)$ is any non-empty subset of $(V)$, then $(W)$ is called a subspace of $(V)$ if $(W)$ itself is a vector space over the field $(R)$ under the same vector addition and scalar multiplication as vector space $(V)$.

Let $(V)$ be a vector space over $(R).$ A non-empty subset $(W)$ of $(V)$ is a subspace of $(V)$ if:

1. $(W\subseteq V)$
2. $(W)$ satisfies the following conditions:
   • Closure under vector addition: For any $(u,v\in W),(u+v\in W).$
   • Closure under scalar multiplication: For any $(a\in R)and(u\in W),(au\in W)$.

Alternatively, a subset (W) of a vector space is called a subspace if$$ (au + bv \in W)\ \ for \ all\ \ (u, v \in W)\ \ and \ \ (a, b \in R).

Note that for a subset to become a subspace, it is necessary that it must contain the zero vector. For example, the set of complex numbers $(\mathbb{C})$ is a vector space over $(\mathbb{R})$, and $(\mathbb{R})$ (the set of all real numbers) is its proper subset. Thus,$(\mathbb{R})$ is a subspace of $(\mathbb{C})$ over $(\mathbb{R})$. --- # References [[Vector Spaces]]