21-06-2024 19:44

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Tags : mathematics linear algebra Linear Equations

Linear Span of a set

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1.4.1 Linear Span of a Set:

• Definition: Let $(S=u_1,u_2,\dots ,u_n)$ be a non-empty subset of a vector space $(V)$. The linear span of (S), denoted by$(L(S))$, is defined as:
  • $[L(S)={\lambda }_1{u}_1+{\lambda }_2{u}_2+\dots +{\lambda }_n{u}_n\mid {\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\in \mathbb{R}]$
• We say that $(S)$ spans the vector space $(V)$ if $(L(S)=V)$. In other words, every vector in $(V)$ can be expressed as a linear combination of elements from $(S)$.

Additional Notes:

• In such case L(S) a is subspace of V in fact it is the smallest subspace containing S.
• If $(L(S))$ is a subspace, it is the smallest subspace containing $(S)$.
• $(L(L(S))=L(S))$

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References