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In Munkres Analysis on Manifolds the author uses the word ''tuple spaces'' to refer to a special sort of vector space. Further down on the same page (page 6) he discusses the linear isomorphism that maps a tuple to a row matrix.

I am greatly confused by this as I do not understand the difference between a tuple and a row matrix and also I do not know what a tuple space is. I do know linear algebra and I do know the definition of a vector space.

Please could someone help me understand this?

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  • $\begingroup$ I have the same question as you. I do not have access to Munkres book at this time. It would be very helpful to me to know exactly what Munkres writes as regards to, ``Further down on the same page (page 6) he discusses the linear isomorphism that maps a tuple to a row matrix.'' Is it possible for you to provide these details? $\endgroup$ – Michael Levy Jun 28 '19 at 16:19
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An $n$-tuple is just a list of $n$ numbers $x_k$ $\>(1\leq k\leq n)$, written as $$(x_1,x_2,\ldots, x_n)\ .\tag{1}$$ Unless you want to do linear algebra with it you can leave it at that.

In linear algebra tuples are used for various purposes and become part of a certain algebraic technique called matrix algebra. Depending on the purpose, it is convenient to arrange a tuple $(1)$ as an $(n\times 1)$-matrix, or column vector, like so: $$\left[\matrix{x_1\cr x_2\cr \vdots \cr x_n}\right]\ ,\tag{2}$$ and sometimes it is more appropriate to arrange it as $(1\times n)$-matrix, or row vector, like so: $$[x_1\ x_2\ \cdots \ x_n]\ .\tag{3}$$ There is no such thing as an isomorphism involved with this; it's just a typographical convenience. (Of course, as you go on in linear algebra you'll understand better which vectors are by their nature column vectors and which ones should be written as row vectors.)

As column vectors $(2)$ take up much space in text they are sometimes written as $[x_1\ x_2\ \cdots \ x_n]'$, or similar. A good rule of thumb is the following: When an $n$-tuple $(1)$ is abbreviated by $x$ then the letter $x$ denotes as well the column vector $(2)$, and one writes $x'$ when the row vector $(3)$ is meant.

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  • $\begingroup$ Thank you. Then I can ignore the paragraph in the book that discusses a linear isomorphism between tuples and vectors? $\endgroup$ – blue Apr 13 '14 at 11:17
  • $\begingroup$ You can ignore "the linear isomorphism that maps a tuple to a row matrix", but you cannot ignore the isomorphism that maps "abstract" vectors to tuples after a basis has been chosen. $\endgroup$ – Christian Blatter Apr 13 '14 at 11:50
  • $\begingroup$ @ChristianBlatter Why can you ignore that there is an isomorphism between the two spaces? Aren't they different vector spaces defined differently but an there is an isomorphism? And could you please explain again in detail the second part of your comment? Thanks a lot. $\endgroup$ – LearningMath Sep 25 '15 at 22:15
  • $\begingroup$ @notorious: Writing $[2\ {-3}\ 5\ 7]$ instead of $(2,{-3},5,7)$ does not exploit an isomorphism; it is just a useful typographical convention. On the other hand it is a fundamental fact of linear algebra that any finitely generated real vector space is isomorphic to the "standard model" ${\mathbb R}^n$ for some $n\geq0$. $\endgroup$ – Christian Blatter Sep 26 '15 at 8:58
  • $\begingroup$ @ChristianBlatter That was what I was asking about. We have the "standard model" or the vector space of n-tuples called $\mathbb{R}^n$. And then, there is another vector space of the row matrices and another one of column matrices. I consider these three as different vector spaces but we can establish an isomorphism between them. Because the notion "vector" is defined as a member of a vector space. The "standard model" is just another vector space in the pool of vector spaces and it happens that its members are n-tuples. Is my understanding right? $\endgroup$ – LearningMath Sep 28 '15 at 7:32
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I think the point of the author is simply saying that you can understand $\mathbb R^n$ as a set of $n$-tuples, but also as a set of $n$-rows or $n$-columns. It doesn't matter how you look at it, all three views are isomorphic.

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There is a formal difference between the tupel space $$ \mathbb R^n:=\mathbb R \times \dots \times \mathbb R $$ and the space of row vectors $\mathbb R^{1,n}$, which is defined as the space 'numbers arranged in a row' (think arranged in a spreadsheet). In the book you linked, the difference is emphasized by using different kinds of brackets: $(x_1,\dots,x_n)$ in the former, $[x_1,\dots,x_n]$ in the latter case.

The spaces are defined differently first. Then one shows that they are isomorphic. This means in terms of linear algebra they are equivalent.

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  • $\begingroup$ (1) I found angle-brackets used to denote ordered pairs of objects (tuples?) and Bourbaki-style structures: $\textbf{Mycategory}=\langle X, A(X), \circ \rangle$. From wikipedia, using angles or parenthesis is a matter of taste, but maybe there is something more that I missed? (2) Are the comma-separated objects in sequent calculus (i.e. $A_1, A_2, \dots, A_n \vdash C_1, C_2, \ldots, C_m$) isomorphic to tuples as well? $\endgroup$ – Niriel May 3 '15 at 17:24

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