Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $V$ be a vector space and $W$ be a subspace of $V$.

Define $I\triangleq \{S\subset W: \text{span} S= W\}$.

Since $\text{span} W=W$, $I$ is nonempty.

Under what conditions, is $I$ finite?

Moreover, i don't understand why subspace $W$ is in the hypothesis, since it seems to me that this hypothesis does not generalize anything more than just saying vector space $V$. Is there anything more general by saying "Let $W$ be a subspace of $V$" rather than not mentioning $W$ in this problem?

share|cite|improve this question
@Tobias To be honest, i cannot think of any approach to attack this problem when $W$ is infinite. Would you give me a hint? – Jj- Feb 14 '13 at 13:55
You're right, $V$ is useless here. – 1015 Feb 14 '13 at 13:58
Hint: If $v$ appears in some spanning set, then you can replace $v$ by $av$ for any non-zero scalar $a$ and not change the fact that the set spans the entire space. – Tobias Kildetoft Feb 14 '13 at 13:59
Consider the cases where $W$ has finite/infinite dimension over the field $K$. Consider $K$ finite/infinite. – 1015 Feb 14 '13 at 13:59
up vote 1 down vote accepted

Hint: Try to prove this (using the comments): $|I|<\infty \iff |W|<\infty$.

($W$ can be infinite either if the base field is infinite or if the dimension is infinite.)

share|cite|improve this answer

If $W$ is finite, how could $I$ be infinite? And if $W$ is infinite, what happens if you just remove a singleton from $W$? How many singletons are there?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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