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?
Best Answer
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.