Question regarding question about hyper planes? I am beginning to study linear algebra study and came across this exercise: A subset of a vector subspace is called a hyperplane or more precisely a linear hyper- plane if it is a proper subspace and such that it, together with one single further vector, generates our whole vector space. Show that: a hyperplane together with any vector not belonging to the given hyperplane generates all of our original vector space. I'm confused as it seems the definition of a hyper plane provides the answer straight away. That is: I have a hyper plane, I have a single further vector, so I must have a generating set. Am I missing something?

Things could have been worded a little differently - seems understandable for you to be missing what you're missing.

Say $V$ is the original vector space and $H$ is our hyperplane. The definition means that there _exists_ a vector $x$ such that $H$ and $x$ generate $V$. The problem asks you to show that $H$ and $x$ generate $V$ for _every_ $x$ (in $V$ but not in $H$).

HINT: Say $V$ is generated by $H$ and $x_0$. Say $x$ is a vector in $V$ which is not in $H$. Now _since_ $H$ and $x_0$ generate $V$ it follows that $x=???$, and then the fact that $x$ is not in $H$ shows that ???