Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

For the following subsets $S$ determine whether $S$ is a subspace of the $F$-vector space $V$. Justify:

  1. Let $F$ = $\mathbb{R}$ and $V$ = $M_n(\mathbb{R})$, the set of $n \times n$ matrices with real entries. Let $A$ be a particular $n \times n$ matrix with real entries.

$$S = \{B\in V | AB \neq BA\}$$

  1. $F$ = $\mathbb{C}$, $V$ = $\mathbb{C}$ and $S$ = $\mathbb{R}$, viewed as a subset of $\mathbb{C}$.
share|cite|improve this question
up vote 4 down vote accepted

For 1, $S$ cannot be subspace because the zero element in $V$, the zero $n\times n$ matrix $O$, does not belong to $S$, because $$AO=O=OA.$$

For 2, $S$ is not a subspace of the $F$-vector space $V$. To see this, let $1\in S=\mathbb{R}$, and $\sqrt{-1}\in F=\mathbb{C}$. However, $$\sqrt{-1}\cdot 1=\sqrt{-1}\not\in S=\mathbb{R}.$$

Note added: However, if $F=\mathbb{R}$, $V=\mathbb{C}$ and $S=\mathbb{R}$, then $S$ is a subspace of the $F$-vector space $V$

share|cite|improve this answer
Thank you so much. That makes so much sense. :D – Kyra Feb 1 '12 at 4:26
@Kyra: You are welcome. – Paul Feb 1 '12 at 4:28

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.