I'm studying for a Linear Algebra exam and found this question that I can't find an answer to.
Let $V$ be a 5-dimensional vector space, and let $X$ be a subset of $V$ which spans $V$. What can be said about $X$?
$a)$ Must be linearly independent
$b)$ Must consist of of at least 5 elements
$c)$ Must have exactly 5 elements
$d)$ Must have at most 5 elements
$e)$ Must be a basis for V
$f)$ Must be linearly dependent
Nothing is said about linear dependence, so I'm comfortable eliminating $a$, $e$, and $f$. After that, I'm not quite sure.
I know that a basis for $V$ contains 5 elements, but I don't know what else can be inferred from the information.
It will be clear if you remember that the (finite) dimension $d$ of a vector space is also characterised by:
(a) $d$ is the minimal number of elements of a spanning subset;
(b) $d$ is the maximal number of elements of a linearly independent subset.