Suppose that $V$ is an n-dimensional vector space over a field $F$ and $\{\vec{v_1}, ...,\vec{v_m}\}$ is a linearly independent set in $V$. How do I show that $m \leq n$ and $\exists \{\vec{v_{m+1}},...\vec{v_n}\} $ such that $\{\vec{v_1}, ...,\vec{v_m},\vec{v_{m+1}},...\vec{v_n}\}$ is a basis for $V$?
The book I'm using gives a hint involving using the basis for $V$ and creating the set of vectors composed of $\{\vec{v_1}, ...,\vec{v_m}\}$ and the basis of $V$, but i don't really understand what it is asking with that.
