Ontological and Notational question concerning Vectors and Matrices

Question

We have a definition of a n-tuple based on Set Theory.

In various text-books regarding Linear Algebra it states that a vector (a "Row Vector") - and here we're talking about a n-tuple, not just a general not-what vector that can be any sort of element in a vector space, that does not have to be a n-tuple - well then, this is just a $1\times n$ matrix. A "Column Vector" would be just a $n \times 1$ matrix.

I have a soft spot regarding definitions, formalizations and ontological questions.

Question: 1) Is an n-tuple in Linear Algebra is regarded really as as a matrix? Whether it be a $1\times n$ matrix or a $n \times 1$ one. Or whether this is a representation of a n-tuple in the matrix world.

2) Is a matrix fundamental entity one declares or is it defined based on other entities? Namely, as a special kind of tuple. I can think of a way of defining this. But I am interested in what is the convention in the mathematical community.

I hope my questions are understood.

Thank's in advance for your answers!

Addition: After some discussion and reflection I can now state that my question is regarding the hierarchy of the mathematical objects. When reading several text-books in linear algebra they repeatedly point out that we can think of a vector as a special kind of matrix, and so we can have two different vectors - row and column vectors. I was asking who is more primitive and who is defined based on the other. Also what is the relation between a vector as $n$-tuple and a row/column vector.

Answer 1

1. You can pretend vectors (rows or columns) are just matrices, and from a mathematical point of view, there is nothing wrong with doing that. All operations continue to be defined just well (sum, addition, products, etc). If you were to formalize linear algebra, it seems like a reasonable course to take; however, as humans, the interpretation of vectors being arrows in space (unlike linear transformations) is probably more intuitive. So yes, it really is (or could be) a matrix but no reason why it must be one.
2. Most objects can be constructed from set theory, including matrices. However most spaces have their own set of axioms and matrices have their own. So if you wanted, you could say it is defined on sets (as a special kind of tuple) or, if you wanted, you could take them as primitive (fundamental entities in your terms) and carry on.

The question on how they are taken to be (defined or primitives) is a question involving philosophy of mathematics (plenitude platonists will take them as primitives, nominalists as defined, and so on). To my knowledge, there is no wide scale survey of this question, but this survey for philosophers of mathematics suggest that most take them as primitives.

Answer 2

The answer to this question really depends on who you ask, and how deep down the rabbit hole you want to go. I definitely can't cover all of the material out there, but hopefully this will provide a starting point.

Edit: This answer has changed in response to feedback from the OP. Check the edit history for previous versions.

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Set Theoretic Definitions of 'Tuple'

In set theory, the principle definition of the "ordered pair" is the Kuratowski definition

An ordered pair is a set $(a,b):=\{\{a\},\{a,b\}\}$

There are different definitions for an $n$-tuple, but the two that I have most commonly encountered both derive from the Kuratowski definition of the ordered pair. The first is that an $n$-tuple is constructed by nesting ordered pairs:

The empty set is a $0$-tuple.

If $x$ is an $n$-tuple, then $(x,y)$ is an $(n+1)$-tuple.

So, for example, the triple $(a,b,c)$ is identified with the set

$$((a,b),c)=\{\{\{a\},\{a,b\}\},\{\{a\},\{a,b\},c\}\}$$

The second definition is that an $n$-tuple is a function from the natural number $n$ (treated as a set, see set-theoretic definitions of natural numbers), to some other set.

An $m\times n$ matrix is an $m$-tuple whose entries are $n$-tuples.

A function from a set $X$ to a set $Y$ is a subset of the Cartesian product $X\times Y$.

If $f$ is a function from $X$ to $Y$, then for all elements $x\in X$, there exists one and only one element $y\in Y$ such that $(x,y)\in f$

For all sets $X$ and $Y$, let $Y^X$ denote the set of functions from $X$ to $Y$

Thus, $X^n$ denotes the set of functions from $n$ to $X$. In this definition, the triple $(a,b,c)$ is identified with the set

$$\{(1,a),(2,b),(3,c)\}=\\ \{\{\{\emptyset\},\{\emptyset,a\}\},\{\{\emptyset,\{\emptyset\}\},\{\{\emptyset,\{\emptyset\}\},b\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},\{\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},c\}\}$$

(you might need to double-check the braces for this one)

In both cases, an $m\times n$ matrix is defined as an $m$-tuple whose entries are $n$-tuples.

Addendum

Using the second definition of "tuple," we may choose also to describe an $m\times n$ matrix as a function $m\times n\to X$ rather than $m\to X^n$.

On the 'Primitiveness' of Different Definitions

Note that while both definitions make use of the ordered pair, they are distinct from one another - that is, they cannot be used interchangeably. Furthermore while these may be the most common definitions that I have seen, they are not any more "correct" than other definitions - and there are many, many, more definitions. In general, set-theoretic representations of algebraic or otherwise "non-set-theoretic" objects stem from an attempt to adapt the theory of those objects (in this case linear algebra) to set theory, not the other way around. In this sense, the $n$-tuple is the primitive object, and the set-theoretic interpretation is a representation of an $n$-tuple in a world of sets.

There is "evidence" for this in the fact that linear algebra predates axiomatic set theory considerably. Furthermore, all formalizations of linear algebra, including ones based on type-theory and other non-set-theoretic foundations, must adhere to the rules of linear algebra - whereas linear algebra need not adhere to any particular formalization. In other words, things like Cramer's rule, Gaussian elimination, and matrix multiplication must be present in every axiomatization of linear algebra, regardless of whether or not these things are readily described by the axiomatic system being used.

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Response to Question 1)

The set-theoretic definitions of "matrix" and "$n$-tuple" do not readily distinguish $n\times 1$ and $1\times n$ matrices.

However, this is not to say that a distinction cannot be made. Depending on how you choose to formalize linear algebra, I see two ways to go about this.

Disclaimer: this is my own work and, as far as I know, there is no canonical set-theoretic definition of "matrix" that actually makes the distinction between $1\times n$ and $n\times 1$ matrices, or which identifies such matrices with $n$-tuples.

Method 1

For a "scalar" domain $X$, let $X^n$ be the set of all $n$-tuples. The set of all matrices over $X$ is defined as...

$$U_X=\bigcup_{m\in \Bbb{N}_1}\bigcup_{n\in\Bbb{N}_1}\left(X^m\right)^n$$

...where $\Bbb{N}_1=\{n\in\Bbb{N}:1\subseteq n\}$. A linear algebra on $X$ is a structure $\langle U_X, +, \cdot, ^\intercal,\cdots\rangle$ satisfying [insert axioms of linear algebra here]. For all natural numbers $n$, an element $v$ of $U_X$ is a $1\times n$ matrix iff $v\in\left(X^1\right)^n$, and an $n\times 1$ matrix iff $v\in\left(X^n\right)^1$. As $X^n$ is not a subset of $U_X$, $n$-tuples are not elements of the linear algebra over $X$.

Method 2

For a "scalar" domain $X$, let $X^n$ be the set of all $n$-tuples. Let $U_0=X^1$, and...

$$U_{n+1}=\bigcup_{i\in\Bbb{N}_1}{U_n}^i\qquad;\qquad U_X=\sup\{U_n:n\in\Bbb{N}\}$$

...where $\Bbb{N}_1=\{n\in\Bbb{N}:1\subseteq n\}$. Define the function $\text{ind}$ as follows:

For any infinite sequence $a$ of natural numbers and element $u\in U_X$, $\text{ind}(u)=a$ iff $u\in\left(\left(\left(\left(U_0\right)^{a_1}\right)^{a_2}\right)^{a_3}\right)^\cdots$

A linear algebra on $X$ is a structure $\langle U_X, +, \cdot, ^\intercal,\cdots,\text{ind}\rangle$ satisfying [insert axioms of linear algebra here]. For all natural numbers $n$, an element $v$ of $U_X$ is a $1\times n$ matrix iff $(\text{ind}(v))_1=1$, $(\text{ind}(v))_2=n$ and for all $i>2$, $(\text{ind}(v))_i=1$. An element $v$ of $U_X$ is an $n\times 1$ matrix iff $(\text{ind}(v))_1=n$, and for all $i>1$, $(\text{ind}(v))_i=1$. Again, no $n$-tuples are present in $U_X$.

This second method has the sole advantage of being able to define arbitrary tensors and hypermatrices.

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In Response to Question 2)

A matrix is indeed a special kind of tuple. Whether or not this makes matrices more or less fundamental depends entirely on how you choose to define "fundamental." From a purely set-theoretic standpoint, a matrix is just a particular kind of set. Outside of this context, matrices may be defined in different ways to make them easier to use. For example, distinguishing row and column vectors (without consideration for any potential set-theoretic definitions), and treating vectors as a type of matrix allows matrix operations to be applied to vectors. This makes certain types of computation considerably easier.

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Summary:

A matrix is type of tuple.

An $n$-tuple whose entries are not tuples is not a matrix.

If a vector is an $n$-tuple whose entries are not tuples, then it is not a matrix (it is neither a $1\times n$ nor an $n\times 1$ matrix).

Both $1\times n$ and $n\times 1$ matrices can be constructed from sets of tuples, neither is more fundamental than the other.

Every formalization of linear algebra must be capable of encoding the same information, so the identification of $n$-tuples with $1\times n$ and $n\times 1$ matrices is generally acceptable even if it conflicts with set-theoretic definitions.

In practice, if a strict distinction between a row and column vector needs to be made, it is best to treat an $n$ dimensional row vector as an $n\times n$ matrix whose only nonzero entries are in the first row, and a column vector as the transpose of a row vector (or vice-versa).

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See also: Direct Product, Categorical Product, Inner and Outer Product, Multilinear Algebra, Tuple, and Function Space

Note: separate notions of "tuple" may be found in alternative set theories, type theory, and category theory. It is also possible to define an order on a set to encode the same.

Answer 3

Short answer in response to the bold really in this part of your question:

1) Is an n-tuple in Linear Algebra is regarded really as as a matrix?

To most working mathematicians (both pure and applied) asking what a mathematical object "really is" is irrelevant, and may not even make sense. What matters is how that object behaves: the axioms that describe its properties and determine the things you can prove using them.

Different examples or instantiations - vectors as tuples in rows or columns, as functions in a function space, or as states of a quantum mechanical system - allow for different insights and uses.

If you're familiar with object oriented programming you will recognize that point of view