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We sometimes right a column vector as $(x_1,x_2,x_3)$ with parentheses (to save space and not confuse it with a row vector), we also tend to write euclidean vectors in this way, in both cases it looks like a 'tuple', we know that $\mathbb R^n$ is defined using tuples, is there an equivalence between the two? Can we write a column vector as being equal to a tuple, perhaps in the column vector case this is just an alternative notation for a column vector and in fact not a tuple?

If there was an equivalence the difficulty is that both a row and column vector could in theory be written as a tuple, and as we know the matrices are not equal.

user3840170
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user37577
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    I think it is just about the notations. In my opinion, when we talk about the "vector," we assume the linear structure. When we talk about "tuple," there is still no operation。 – VicaYang Jul 16 '22 at 08:35
  • Some textbooks show a column vector as a tuple but it shouldn't cause any confusion in matrix multiplications. For example if a $4\times 3$ matrix $A$ is multiplied by a vector $v$ as $Av$ it should be clear that the dimensions of $v$ is $3\times 1$ and so it a column vector even if it's written as $v=(x_1,x_2,x_3)$. But most of the textbooks use transpose $v^{T}$ to change a row vector to a column vector so instead of $Av$ they write $Av^{T}$. Secondly, $\mathbb{R}^n$ is defined as tuples and remember that each point in $\mathbb{R}^n$ corresponds to a vector and vice versa. – Dan Jul 16 '22 at 08:35
  • @ VicaYang. If we look at $\mathbb{R}^{n}$ as the space of all $n$-tuples, it is a vector space with the usual addition and multiplication by scalars on $n$-tuples. – Medo Jul 16 '22 at 08:37
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    @Medo - The key point there is *with". There is no disputing that $\mathbb{R}^n$ is a vector space, when given those operations. One might be speaking of tuples with or without consideration of that vector space structure: one may be simply speaking of the set, not the vector space. It's one of those things that really just depends on the context and the writer's intentions. – PrincessEev Jul 16 '22 at 08:39
  • @Eevee Trainer. Yes. And I think this is the answer: "It depends on the context". Please see my answer below and comment ... – Medo Jul 16 '22 at 08:41
  • Welcome to Math.SE! <> Because you're potentially interested in physics, it's worth emphasizing that "vector" means different things to mathematicians than to physicists. In mathematics, a vector is usually construed as an element of a vector space (a set and two operations satisfying axioms). To a physicist, a vector generally refers to a quantity associated to a coordinate system that transforms in a particular way under change of coordinates. (The physicists' transformation law is motivated by the mathematical concept of matrix multiplication.) – Andrew D. Hwang Jul 16 '22 at 19:49

2 Answers2

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A vector is, definitionally, an element of a vector space; they may be tuples, but don't have to be. (Functions can form a vector space, for instance.)

Critically, a vector space requires additional structure: one needs to define an addition and scalar multiplication, and the scalars and tuple entries must come from a field. If we define the operations in the usual pointwise fashion, then, sure, all tuples of a particular fixed size are a vector space.

But as stated, tuples by themselves are not vectors; simply tuples.

The choice of a row vector versus a column vector is not particularly noteworthy and important, and the operation of transposition lets one easily slide from one perspective to the other. It's only really important for the sake of matrix operations and certain properties associated with matrices.

I would hesitate to say there's any sort of "equivalence" going on here, however, prior to some definition of said equivalence being given. Tuples are just natural examples of vectors, but not equivalent to them. Row vectors and column vectors are not the same thing, but concerns focused on one are often easily translated to the other.

PrincessEev
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    One thin that slightly baffles me about the whole thing is $R^n$ is given as a set of n-tuples yet a euclidean vector of dimension $n$ is given as an element of $R^n$. – user37577 Jul 17 '22 at 10:16
  • @user37577 An euclidean vector of dimension $n$ is an element of an $n$-dimensional real inner product space (an euclidean vector space). Choosing a basis allows us to identify each vector with an $n$-tuple. – Filippo Jul 22 '22 at 17:34
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They are essentially the same!

A point $p=(x_{1},\dots,x_{n})$ in $\mathbb{R}^{n}$ is the head of the vector $v$ that starts at the origin and ends at the point $p$. The vector $v$ can be represented by the coulumn $v=[x_{1},\dots,x_{n}]^{T}=\begin{pmatrix} x_{1} \\ \vdots \\ x_{n} \end{pmatrix}$. It can also be identified with its head, the point $p$. One needs to choose the representation suitable for the context.

Medo
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