So I've seen two definitions of this:

Let $V$ be a vector space with subspace $W$. We say that $X \subseteq V$ spans $W$ if and only if

(Definition 1): Every $\vec{w} \in W$ can be written as a linear combination of vectors in $X$.

(Definition 2): $span(X) = W$.

I don't think the two definitions are equivalent are they? Clearly the condition in Def. 2 implies the condition in Def. 1, but not the other way round.

For example take $V$ to be the vector space of ordered pairs over $\mathbb{Z}_2$, then if $W = \{(0,0),(1,1)\}$ and $X=\{(0,1),(1,0)\}$, $X$ spans $W$ according to the first definition, but not the second.

What then would you have understood if I were to tell you that "$X$ spans $W$"? Would you have agreed or disagreed with me?

I know the usual drill is "clarify your definitions from the start" in whatever work you're doing, and that as long as you do that it will be fine, but I'm just wanting to know what peoples' immediate interpretation would be. Because it does impact a little on the reading and understanding process.

(For the record I'm a fan of Definition 1.)

  • 3
    $\begingroup$ One usually understands $X$ spans $W$ to include the condition that $X$ is a subset of $W$. $\endgroup$ – Gerry Myerson Oct 17 '12 at 1:22
  • $\begingroup$ I would vote for definition 2. $\endgroup$ – Bitwise Oct 17 '12 at 1:57

My feeling for this terminology is that the phrase "$X$ spans $W$" should be consistent with the phrase "the span of $X$ is $W$". It is standard to define "the span" of a set, and so the first definition, which implies that $X$ could span distinct subspaces, is inconsistent with "the span".

I find this an interesting question actually because certain phrasings of the definition of "span" could lead to this confusion; namely the phrasings involving spanning of subspaces.

Probably a good definition is: if $V$ is a vector space and $X\subseteq V$ is a set then we say that $X$ spans $V$ if every $v\in V$ can be written as a linear combination of elements of $X$. This way you have to refer to a set of vectors within a vector space.

So in this case to say that $X$ spans $W$ only makes sense in the case that $X\subseteq W$. It is as if $V$ ceased to exist for a brief moment of mathematical thought.

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