0
$\begingroup$

If the dimension of the span of a subset $X$ is equal to the dimension of the vector space $V$, is $X$ a spanning set of $V$?

$\endgroup$
  • 1
    $\begingroup$ If the dimensions are finite then yes. The reason is that any large enough set of linearly independent vectors form a basis. $\endgroup$ – Zeekless Jun 1 at 9:00
2
$\begingroup$

In a space $V$ of dimension $n$ the only subspace $X$ of dimension $n$ is $V$ itself. Indeed, if there is a vector $v$ which is not in this subspace then the span of $X\cup \{v\}$ would have dimension greater than the dimension of $V$ which is impossible. Hence $X$ must be equal to $V$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.