Union of Two Subspaces

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If $U \subset W$ (or vice versa) then $U \cup W = W $ so if W is a subspace so is the union.

Otherwise there is $a \in U ,\ a \notin W$ and $b \in W,\ b \notin U$

So, $a, b \in U \cup W$.

But $a + b \notin U$ otherwise with $(-a) \in U$ then $(-a) + a + b \in U \implies b \in U$

And similarly $a + b \notin W$

Therefore $a + b \notin U \cup W$, i.e. there are vectors $a, b$ in $U \cup W$ whose linear combination is not in $U \cup W$, so $U \cup W$ is not closed.

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Sabih Sai
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Updated on August 19, 2022

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  • Sabih Sai
    Sabih Sai about 1 year

    Q. Say $U$ and $W$ are subspaces of a a finite dimensional vector space $V$ (over the field of real numbers). Let $S$ be the set-theoretical union of $U$ and $W$. Which of the following statements is true:

    a) Set $S$ is always a subspace of $V$

    b) Set $S$ is never a subspace of $V$

    c) Set $S$ is a subspace of $V$ if and only if $U=W$

    d) None of the above

    I've researched on this question and found a proof that says set $S$ is not always subspace of $V$ (Is it true?).

    If $U=W$ then $S$ is a subspace of $V$ since $U$ and $W$ are closed under addition hence $U \cup W = U$ (or $W$). So can I say that $S$ is a subspace of $V$ if and only if $U = W$?

    Also, please can you give an example of how to calculate union of two vector spaces? So what would be the union of vector spaces $\begin{pmatrix} a& 0 &0\end{pmatrix} \cup \begin{pmatrix}0& b& 0\end{pmatrix}$ ?

    Thanks

    • Anubhav Mukherjee
      Anubhav Mukherjee about 8 years
      It is true if the underline field is $\mathbb{R}$... as a consequence of Baire category theory... other wise it may not be true...for an example if you consider the underline field as $|mathbb{Z_2}$ then you can find some counter example
    • André Nicolas
      André Nicolas about 8 years
      The union of $U$ and $W$ is a subspace of $V$ if and only if either $U\subseteq W$ or $W\subseteq U$. For a proof, suppose that neither subset relation holds. Let $x$ be in $U$ but not in $W$ and let $y$ be in $W$ but not in $U$. Show that $x+y$ is not in the union of $U$ and $W$.
    • Anubhav Mukherjee
      Anubhav Mukherjee about 8 years
      have a look in this link mathoverflow.net/questions/26/…
  • Sabih Sai
    Sabih Sai about 8 years
    Thank you very much, it makes sense!