1

Let $V_1,V_2$ are subspaces of vector space $V$ . Differences of $V_1 \cup V_2$ and $V_1 +V_2$ ?

Road Human
  • 1,012
  • 3
    @Najib: Poor selection, NOT a duplicate of your linked post. Please read questions before jumping the gun. (Don't be so quick to close questions you haven't bothered to read, or using a linked post you think addresses the question, but of which you haven't actually read and compared.) This post is asking for the difference between the union of two subspaces and their sum. Nowhere do I see any question related to whether $V_1\cup V_2$ is a subspace. – amWhy Jan 12 '15 at 14:38
  • 3
    @amWhy By definition, $V_1 + V_2$ is the smallest subspace of $V$ containing $V_1 \cup V_2$. So if $V_1 \cup V_2$ is not a subspace... I did read the post, thank you very much... I do assume the OP knows the most basic definitions though. Maybe that's asking too much from your point of view? – Najib Idrissi Jan 12 '15 at 14:40

2 Answers2

7

Let $V = \Bbb R^2,V_1 = \{(x,0)\mid x \in \Bbb R\}$ and $V_2 = \{(0,y)\mid y \in \Bbb R\}$. Then $$V_1 + V_2 = \big\{(x,y)=(x,0)+(0,y)\in \Bbb R^2 \mid (x,0) \in V_1 \text{ and }(0,y)\in V_2\big\}=\Bbb R^2$$ but $$V_1 \cup V_2 =\{(a,b)\in \Bbb R^2\mid a=0 \text{ or } b = 0\} \neq \Bbb R^2$$

Surb
  • 55,662
4

The first is in general not a subspace and is the set of vectors that are in $V_1$ or in $V_2$. The second is the set of sums $v_1 + v_2$ where $v_i \ in V_i$ and is a subspace. It is by the way generated as vector space by the first one.

Olórin
  • 12,040