Basis

==2024-12-11

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wow

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Theory

Given a Vector Space $V_n$ defined over a Mathematical Field $\mathbb{F}$, If for a Subspace of a Vector Space of $V_n$ say $A$, the span of the subset of $A$, span$(H)=A$ then H is the basis of A.

Formal Defination

Let $V$ be a Vector Space over a field $\mathbb{F}$, a Subset $B\subset V$ is called a basis for $V$ if:

1. The vectors in B are Linearly Independent.
2. The vectors in $B$ Span the vector space $V$. i.e $\mathrm{span}(B)=V$

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PTR