Difference between union of 2 vector spaces and the direct sum of them

**arbolis** · February 20th 2009, 11:24 AM

What's the difference between $V \cup W$ and $V \oplus W$ , $V$ and $W$ are vector spaces. I didn't find the answer on the Internet.
A friend told me that $V+W=$ $\text {span} (V \cup W)$ . But then if $V$ and $W$ are $\mathbb{R}^2$ as vector space over $\mathbb{R}$ then I get that $\mathbb{R}^2+\mathbb{R}^2$ is formed by ... ahh well I'm unsure of everything.

**HallsofIvy** · February 20th 2009, 12:00 PM

Originally Posted by arbolis

What's the difference between $V \cup W$ and $V \oplus W$ , $V$ and $W$ are vector spaces. I didn't find the answer on the Internet.
A friend told me that $V+W=$ $\text {span} (V \cup W)$ . But then if $V$ and $W$ are $\mathbb{R}^2$ as vector space over $\mathbb{R}$ then I get that $\mathbb{R}^2+\mathbb{R}^2$ is formed by ... ahh well I'm unsure of everything.

Well, yes, $R^2\cup R^2= R^2$ and since $R^2$ is already a subspace its span is also just $R^2$ . That's not a very good example!

Suppose V= {(x, 0, 0)} in $R^3$ and W= {(0, y, 0)} in $R^3$ . Then $U\cup V$ is the set of all vectors of the form (x, 0, 0) or (0, y, 0). Geometrically, that is the x-axis and the y-axis but NOT the points in between. That is not a subspace because, while it contains (1, 0, 0) and (0, 1, 0), it does NOT contain their sum, (1, 1, 0). The direct sum of V and W is the xy-plane: {(x, y, 0)}} which includes (1, 0, 0), (0, 1, 0) and (1, 1, 0).

**kalagota** · February 20th 2009, 08:04 PM

in general, an element of $V\cup W$ is any element of $V$ or of $W$ . however, an element of $V \oplus W$ is a vector in which you can write it as a sum of an element of $V$ and of $W$ . indeed, $V \oplus W = \mbox{span }{V\cup W}$

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