For instance we have v1 = i, v2 = j, v3 = k unit vectors. Can we say that v1,v2 and v3 span the R^2 or is saying that wrong? I mean, I know that R^3 can be represented as the linear combination of these 3 vectors but in the book for definition it says v1,v2,v3 ... ,vk span the vector space V provided that every vector in V is a linear combination of these k vectors. So in my example in R^2 every vector can be represented as the linear combination of i,j,k so if I say i,j,k span R^2 is it correct or wrong since I didn't say R^3?
The meaning of span for vectors
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But it is conceptual question. It just came yo my mind.
HallsofIvy
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Yes, we can say that i, j, and k span R^2 because any vector in R^2 can be written xi+ y+ 0k for some numbers x and y. It is a bit of overkill because you don't really need "k".
Dr.Peterson
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I would say it's wrong, though I can see other arguments.
The trouble as I see it is that i, j, and k are not in R^2! That is a set of ordered pairs, not ordered triples, which i, j, and k are, as you are implicitly taking them. In order to say what you are saying, you have to think of R^2 as a subspace of R^3, which is not what it inherently is.
Cubist
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I think the following two statements might be different:-
1. The set of vectors S span the space V
2. The span of S is V
Statement 1 is used in the OP. It's a statement about two inputs S and V that implies a certain property of S and V. And given the definition provided in the original post it seems OK to say that i,j,k span R^2 (or R^3, or R^1)
Statement 2 has one input S and a single output V. For example, the span of i,j,k is R^3. (Here you could not say R^2).
However I'm not sure. Hopefully others will add to this thread.
1. The set of vectors S span the space V
2. The span of S is V
Statement 1 is used in the OP. It's a statement about two inputs S and V that implies a certain property of S and V. And given the definition provided in the original post it seems OK to say that i,j,k span R^2 (or R^3, or R^1)
Statement 2 has one input S and a single output V. For example, the span of i,j,k is R^3. (Here you could not say R^2).
However I'm not sure. Hopefully others will add to this thread.
ISTER_REG
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Let me throw in a little example. Let's say we have the vectors [MATH]v_1 = \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix}[/MATH] and [MATH]v_2 = \begin{pmatrix} 1 \\ 6 \\ 0 \end{pmatrix}[/MATH] , question do these span the [MATH]\mathbb{R}^2[/MATH]? The answer to this is, no, because the vectors have three components and are therefore in [MATH]\mathbb{R}^3[/MATH]. What you can say at most is that these vectors span a subspace of [MATH]\mathbb{R}^3[/MATH] (which is a plane), namely the area of the [MATH]X,Y[/MATH] axis, if you think of it in three dimensions. But this is then still no [MATH]\mathbb{R}^2[/MATH]. But it "acts" like the [MATH]\mathbb{R}^2[/MATH]...
Important side note: The vector space [MATH]\mathbb{R}^2[/MATH] is not a subspace of [MATH]\mathbb{R}^3[/MATH] because [MATH]\mathbb{R}^2[/MATH] is not even a subset of [MATH]\mathbb{R}^3[/MATH], see this by cointing the components of the vectors.
Important side note: The vector space [MATH]\mathbb{R}^2[/MATH] is not a subspace of [MATH]\mathbb{R}^3[/MATH] because [MATH]\mathbb{R}^2[/MATH] is not even a subset of [MATH]\mathbb{R}^3[/MATH], see this by cointing the components of the vectors.
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Dr.Peterson
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I agree with this, except for the fact that R^2 is not a subset of R^3. The vectors i, j, k do span the xy-plane within R^3; but that is not the same thing as R^2, even though it looks an awful lot like it ... . It is very likely that the OP really meant only this, but that is not what the question literally says.
Ultimately, this comes down to the definition of R^2, which is where I am sympathetic to the other perspective.
Cubist
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I've done some extra reading, and I now agree with @Dr.Peterson , @Jomo , and @ISTER_REG .
It seems the span "operator" accepts a set of vectors in a space (all input vectors must be of the same n dimensions), and returns an (often infinite) set of vectors (also in n dimensions) that can be reached from all the linear combinations of the input vectors.
EDIT: So I guess a better use of span would be to say i+j+k is not an element of the span of { i+j+0k, i+0j+0k }. However it's not very useful to talk of dimensions in regard to spans.
It seems the span "operator" accepts a set of vectors in a space (all input vectors must be of the same n dimensions), and returns an (often infinite) set of vectors (also in n dimensions) that can be reached from all the linear combinations of the input vectors.
EDIT: So I guess a better use of span would be to say i+j+k is not an element of the span of { i+j+0k, i+0j+0k }. However it's not very useful to talk of dimensions in regard to spans.
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@Dr.Peterson @Cubist @ISTER_REG @Jomo Thank you so much for all of your help.
1 - So as I understand i = (1,0,0), j = (0,1,0), k = (0,0,1) span xy plane in space having 0 component for its z but we can not say i,j,k span R^2 since R^2 is not a subspace of R^3.
2 - Saying that span(i,j,k) = xy plane in space is wrong as I understand from your answers. I should say span(i,j,k) = R^3 here because it should result in one output.
However, when I tried to figure out this concept I also found this in wikipedia "Given a vector space V over a field K, the span of a set S of vectors (not necessarily infinite) is defined to be the intersection W of all subspaces of V that contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W, and we say that S spans W.". So according to this definition we can not also say i,j,k span xy plane since it does not include (0,0,1) vector in itself. Therefore, I am confused now. Could you also explain your ideas about this? (I know what I wrote as definition at the beginning was different but it was also from the book "Differential Equations and Linear Algebra" Third Edition published by Pearson written by Henry Edwards, David Penney)
1 - So as I understand i = (1,0,0), j = (0,1,0), k = (0,0,1) span xy plane in space having 0 component for its z but we can not say i,j,k span R^2 since R^2 is not a subspace of R^3.
2 - Saying that span(i,j,k) = xy plane in space is wrong as I understand from your answers. I should say span(i,j,k) = R^3 here because it should result in one output.
However, when I tried to figure out this concept I also found this in wikipedia "Given a vector space V over a field K, the span of a set S of vectors (not necessarily infinite) is defined to be the intersection W of all subspaces of V that contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W, and we say that S spans W.". So according to this definition we can not also say i,j,k span xy plane since it does not include (0,0,1) vector in itself. Therefore, I am confused now. Could you also explain your ideas about this? (I know what I wrote as definition at the beginning was different but it was also from the book "Differential Equations and Linear Algebra" Third Edition published by Pearson written by Henry Edwards, David Penney)
ISTER_REG
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The vectors (1,0,0)^T , (0,1,0)^T and (0,0,1)^T span first of all the [MATH]\mathbb{R}^3[/MATH]. If we say that their z-component is 0, then the system reduces to (1,0,0)^T , (0,1,0)^T respectively (0,0,0)^T.
Yes!
Yes, thats because we have vecors with three components (x,y,z) and even more the [MATH]\mathbb{R}^2[/MATH] is not a subspace of [MATH]\mathbb{R}^3[/MATH], since it is not even a subset of [MATH]\mathbb{R}^3[/MATH]
I think you are confused by the word subset here? So we have from the definition:
[MATH]V[/MATH] is a vector space over a field [MATH]K[/MATH] and [MATH]S \subset V[/MATH] is a subspace of this vectorspace, therefore we have [MATH]\text{span}(S) = \left(\sum_{i=1}^{k} \lambda_i v_i | k \in \mathbb{N}, v_i \in S, \lambda_i \in K\right)[/MATH], the span is the set of all finite linear combinations of the [MATH]v_i[/MATH].
A better idea might be to think of the XY plane, the span then contains all possible vectors that can lie in that plane (all possibilities is more or less the span). For example, just imagine your desk, now you can place a pen on it in almost infinite ways. And that's roughly how you can imagine the span. Now you can ask yourself why this is important and what this definition is supposed to do, but at the latest with the concept of the base it is then completely clear why the span is so important. At least this is how I like to think of the span.
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