basis of a vector space

Topics

• linear algebra

A basis of $\mathbb{V}$ is a set of linearly independent vectors that span $\mathbb{V}$. The number of vectors in a basis of a vector space $\mathbb{V}$ equals to $\text{dim}(\mathbb{V})$. Mathematically, let $v_1,v_2,\dots ,v_n\in \mathbb{V}$ be $n$ linearly independent vectors. Then, they are said to be a basis for $\mathbb{V}$ iff, for any $v\in \mathbb{V}$, there exists $n$ scalars/coefficients ${\beta }_1,{\beta }_2,\dots ,{\beta }_n$ such that,

$$v={\beta }_1{v}_1+{\beta }_2{v}_2+\dots +{\beta }_n{v}_n$$

A nice corollary to this is that the representation of a vector in terms of a basis is unique.

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