Let $V = V_1 \oplus V_2$. Also, $V$ equals the direct sum of $V_3$ and $V_4$.
$V_1$, $V_2$, $V_3$, $V_4$ are subspaces of $V$.

Prove or disprove that:

$$ V = (V_1 \cap V_3) \oplus (V_1 \cap V_4) \oplus (V_2 \cap V_3) \oplus (V_2 \cap V_4) $$

Not sure how to start on this one. I know that the intersection of $V_1$ and $V_2$ is the zero vector, likewise for $V_3$ and $V_4$. Not sure how this helps.

  • 1
    $\begingroup$ Voting down because the OP made no effort to format the question properly. $\endgroup$ – Rhys Sep 4 '13 at 13:19
  • $\begingroup$ @Rhys, he has gained 36 rep, and as many new users, he's probably not used to $\LaTeX$. The post is OK now, are you voting for it or for the user's behavior? +1 because the question is OK, there was a lillte effort in finding out an answer, and this question deserves more than -3 votes. $\endgroup$ – JMCF125 Sep 4 '13 at 15:36

Consider $V = \mathbb{R}^2$.

Let $V_1 = \mathbb{R}\{(1,0)\}$, $V_2 = \mathbb{R}\{(0,1)\}$, $V_3 = \mathbb{R}\{(1,1)\}$, and $V_4 = \mathbb{R}\{(1,2)\}$.

Then $V = V_1 \oplus V_2$ and $V = V_3 \oplus V_4$. However $V_1 \cap V_3 = V_1 \cap V_4 = V_2 \cap V_3 = V_2 \cap V_4 = \{(0,0)\}$.

  • $\begingroup$ I don't see how V1 DirectSum V2 = V. It seems that it just equals (1,1). $\endgroup$ – larry Sep 4 '13 at 13:10
  • $\begingroup$ @user9915 $\mathbb{R}\{(1,0)\}$ is basic vector space notation. $V_1 = \mathbb{R}\{(1,0)\} = \{(r,0) : r \in \mathbb{R}\}$. $\endgroup$ – William Sep 4 '13 at 13:13
  • $\begingroup$ @user9915 $V_1 \oplus V_2$ is the direct sum of two one dimensional vector space: the line spanned by $(1,0)$ and the line spanned by $(0,1)$. This is $\mathbb{R}^2$. $\endgroup$ – William Sep 4 '13 at 13:15

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