Generators of Vector Spaces

What is it?

In Linear Algebra, a vector space $\mathbf{V}$ has infinite elements and, mostly, all vectors of $\mathbf{V}$ can be obtained from a finite set of vectors called $\mathbf{S}$. The set $\mathbf{S}=\{{\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n\}$ which generates the vector space $\mathbf{V}$, is called the generator of $\mathbf{V}$.

All vectors of $\mathbf{V}$ can be obtained from a linear combination of the vectors in $\mathbf{S}$, which makes $\mathbf{S}$ the generator of $\mathbf{V}$, represented by $\mathbf{V}=[\mathbf{S}]$.

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Checking if $\mathbf{S}$ generates $\mathbf{V}$

We can check if a set $\mathbf{S}$ generates $\mathbf{V}$, considering any vector $\mathbf{v}$ of $\mathbf{V}$, and checking if $\mathbf{v}$ is a linear combination of $\mathbf{S}$. In case $\mathbf{v}$ is not a linear combination, then $\mathbf{S}$ does not generate $\mathbf{V}$.