Let $\left\{\boldsymbol v_{1}, \ldots, \boldsymbol v_{n}\right\}$ be a set of generators of a [[vector space]] $V$. Let $\left\{\boldsymbol v_{1}, \ldots, \boldsymbol v_{r}\right\}$ be a [[maximal subset of independent vectors]]. Then $\left\{\boldsymbol v_{1}, \ldots, \boldsymbol v_{r}\right\}$ is a [[basis for vector space|basis]] of $V$. Any vector written as combinations of $\left\{\boldsymbol v_{1}, \ldots, \boldsymbol v_{n}\right\}$ can be rewritten using $\left\{\boldsymbol v_{1}, \ldots, \boldsymbol v_{r}\right\}$ This number $r$, the number of vectors, is the [[dimension of a vector space]] #### Proof Since $\left\{\boldsymbol v_{1}, \ldots, \boldsymbol v_{r}\right\}$ is maximal, $\underset{\text{not zero because independent}}{\underbrace{x_{1} \boldsymbol v_{1}+\cdots+x_{r} \boldsymbol v_{r}}}+\underset{\underset{\text{also non-zero}}{\uparrow}}{y} \boldsymbol v_{i}=0$ for $i>r$. $\boldsymbol v_i$ can be written as a [[linear combination]] of $\left\{\boldsymbol v_{1}, \ldots, \boldsymbol v_{r}\right\}$, $\boldsymbol v_{i}=\frac{x_{1}}{-y} \boldsymbol v_{1}+\cdots+\frac{x_{r}}{-y} \boldsymbol v_{r}$ so any element of $V$ can be generated from this set of vectors. Therefore, it forms a [[basis for vector space|basis]]