A different way to express the span

  • #1
Let us assume that d is a vector in the vector space ℝ2 , then is:

{td | t ∈ ℝ} the same as span{d} ?

Thank you.
 

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  • #2
mathwonk
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yes. but you should look at a definition of "span" and check this yourself.
 
  • #3
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Let us assume that d is a vector in the vector space ℝ2 , then is:

{td | t ∈ ℝ} the same as span{d} ?

Thank you.
Basically, yes. It would generally be better to speak of a linear span instead of just span, but this is a common sloppiness. Span cannot be recommended. Also in such a general context like here, it would be better to add the scalar field as an index ##\operatorname{lin}_\mathbb{R}\{d\}=\operatorname{span}_\mathbb{R}\{d\}##. Span as operatorname is no protected abbreviation, because it is context sensitive. So with even less effort one can write ##\mathbb{R}\cdot d## or ##\sum_{i\in I}\mathbb{R}d_i## if more than one vector is involved.
 
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  • #4
WWGD
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As per @mathwonk 's suggestion, maybe we can use that the span of a set S is the smallest vector space containing the set S. Now, subspaces must preserve *.....


* Referring to operations defined on vector spaces.
 
  • #5
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Basically, yes. It would generally be better to speak of a linear span instead of just span, but this is a common sloppiness.
I don't think it's necessary to include "linear" with "span" since span is already defined in most textbooks to be the set of all linear combinations of a set of vectors. Also, unless we're talking about a completely arbitrary vector space, with neither the dimension nor underlying field known, we usually have some idea about the dimension of the vector space and the field from which the scalars are drawn.
Also in such a general context like here, it would be better to add the scalar field as an index
In the example of the OP in this thread, the scalar field is clearly R.
Let us assume that d is a vector in the vector space ℝ2 , then is:

{td | t ∈ ℝ} the same as span{d} ?
Span(d) is unambiguous in my opinion.
 
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  • #6
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Span(d) is unambiguous in my opinion.
Maybe, but it takes not much to be precise. Especially the scalar field is important. Outside physics it is not automatically clear that it is of characteristic zero or algebraically closed. And even in physics, there is a major difference between ##\mathbb{R}## and ##\mathbb{C}##. E.g. someone recently asked about the span of ##\{z,\bar{z}\}##, in which case you get two different spaces, depending on whether it is a real or complex vector space. It is correct that span is usually the linear span and others are called generated by. I have learnt to call the span of a set ##S## the linear hull of ##S##, which is in my opinion the better term. As our readers are often students, I can't find something wrong about it, to advise them to be as precise as possible. Far too many meaningless discussions take place just because of a different understanding of default or a hidden I meant. In the given example, of course, the question: Is ##\mathbb{R}d=\operatorname{span}\{d\}##? only allows a real vector space, but that is what I wanted to say: ##\operatorname{span}## is context sensitive.
 
  • #7
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Maybe, but it takes not much to be precise.
To me, writing "linear span" instead of "span" seems redundant. Two of the linear algebra books I pulled from my shelf talk only about "span" and don't further qualify it by adding "linear."

If the context is crystal clear, I don't see any advantage in being over-precise.

Especially the scalar field is important.
Sure, but the way the question was written, it was obvious what the scalar field was.

In the given example, of course, the question: Is ##\mathbb{R}d=\operatorname{span}\{d\}##? only allows a real vector space, but that is what I wanted to say: span is context sensitive.
Of course. And when that context is very clear, I don't see any lack of clarity by omitting redundant details. There are some men who alway use a belt and suspenders to hold their pants up.
 
  • #8
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One more thing. Here's the OP again.
Let us assume that d is a vector in the vector space ℝ2 , then is:
{td | t ∈ ℝ} the same as span{d} ?
Just to be clear, the OP has identified the vector space: ##\mathbb {R^2}## and the scalar field: ##\mathbb R##. I would be willing to bet that the response of ##\mathbb R d = span\{d\}## would be confusing to the OP, rather than clarifying.
 
  • #9
WWGD
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I don't think it's necessary to include "linear" with "span" since span is already defined in most textbooks to be the set of all linear combinations of a set of vectors. .

I have always found the equivalent definition that for any set S of vectors, Span(S) is the smallest vector space containing the set S -- the largest of course, being the whole ambient space, which will happen if S contains a basis for S.
 

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