Basis (linear algebra)

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Definition

Let $(V,\mathbb{F})$ be a vector space and let $(b_i)_{i\eq 1}^n\subseteq V$ be a finite set of vectors of $V$. Then $(b_i)_{i\eq 1}^n$ is a basis for $V$ if:

• $\{b_1,\ldots,b_n\}$ is a linearly independent set (over $\mathbb{F}$), and
• $\text{Span}$$(b_1,\ldots,b_n)\eq V$
  • That is to say every $v\in V$ is a linear combination of elements of $\{b_i\}_{i\eq 1}^n$, that is $v\eq\sum_{j\eq 1}^k \lambda_jb_{\alpha_j}$ where $(\alpha_j)_{j\eq 1}^k$ are indices for basis elements and $(\lambda_j)_{j\eq 1}^k\subseteq\mathbb{F}$ are scalars.

Caveat:The infinite case - we're actually able to write $\sum_{\alpha\in I}\lambda_\alpha x_\alpha$ for an arbitrary set of vectors $\{x_\alpha\}_{\alpha\in I}\subseteq V$ for a linear combination, provided that $\vert\{\lambda_\alpha\in\mathbb{F}\ \vert\ \alpha\in I\wedge \lambda_\alpha\neq 0\}\vert\in\mathbb{N}$ - that is only finitely many of the $\lambda_\alpha$ are non-zero, this way all the (possibly infinitely many) zero terms vanish.

• - See Hamal basis