New answers tagged

2

The most common way to express this is that component $c_1$ is not simply connected (it has a hole). Of course, this doesn't help to give a term for $c_2$, but in analogy with political maps, we might call it an enclave.


2

You could say that a digraph has this property if and only if all connected components are strongly connected. I'm not sure if there is one word for it.


6

Because the quotient of $V \times W$ by $W$ is $V$. (more precisely, the quotient of $V \times W$ by the subspace $\{ 0 \} \times W$ is naturally isomorphic to $V$) There are more general situations, but that it covers this particular situation is, in my opinion, a pretty strong motivation. The idea is even clearer in the case of abelian groups; finite ...


1

Two possible motivations: For finite-dimensional vector spaces over finite fields, $$ |V/W|=|V|/|W|\;. $$ And for $m\mid n$, $$ n\mathbb Z/m\mathbb Z\sim(n/m)\mathbb Z\;. $$ More generally, see also the Wikipedia article on equivalence classes. Whenever the equivalence classes of an equivalence relation all have the same size $d$, forming the quotient ...


1

Your lists includes some subjects, some algebraic structures, and some more specific objects. I'll separate them to organize the list a little better. Anyone can feel free to contribute to this list as I'm certainly not an expert in all of this. Subjects: Linear Algebra: the study of finite dimensional vector spaces and the linear transformations ...


1

According to Wikipedia and some papers: Exterior algebra = Grassmann algebra (= differential forms, since they are a construction of the exterior algebra) (=derivations, since derivations are just one possible construction of the dual object to differential forms). Multilinear algebra contains differential forms as a special case, so exterior=Grassmann ...


7

This is a falling factorial: $$ (n)_k = n^{\underline k} = \underbrace{n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)}_{k\text{ factors}} = \frac{n!}{(n-k)!} $$


7

Do we ever put functions as entries of a matrix? Yes. There have been quite a few fantastic examples given, but I'm not sure they get to the heart of your question. If so, are these matrices used in linear algebra or do they have some other special use? The key is not that matrices with functions (or functionals or operators or vectors or matrices ...


0

With reference to a curve $C$ with continuously turning tangent in a metrical space of any number of dimensions, we define a tube as the locus of points at a fixed distance $\theta$, called the radius, from $C$, the distance being measured in each case along a geodesic perpendicular to $C$. HOTELLING, Harold. Tubes and spheres in n-spaces, and a ...


2

$\require{cancel}$ Generally speaking, if we have two relations $R$ and $S$, then we define $$a\,\cancel {R\,}\,b\iff \lnot(a\,R\,b)\\ a\,{}^R_S\,b\iff (a\,R\, b) \vee(a\,S\,b).\tag{*}\label{moi}$$ There are some exeptions, which I personally try to avoid, like $A\subsetneqq B$. Before I go on, I must point out a crucial inconsistency in the OP. In ...


2

I believe in this case, the proper mathematical term for the region would be tube (or tube domain, depending on your focus), so the corresponding surface would be the surface of the tube. While pipe is not (to my knowledge) a formal mathematics term, it is used as a physical example in enough problems (both real and created) that the meaning would be just ...


7

Even just the regular first derivative of a function $$ f:\mathbb{R}^n\rightarrow \mathbb{R}^m$$ Is a matrix of functions, if you define $f$ by its $m$ component scalar functions from $\mathbb{R}^n\rightarrow \mathbb{R}$, or as $$f(\vec{x})=(f_1(\vec{x}),....,f_m(\vec{x})) $$ with $$ \vec{x}=(x_1,....,x_n)$$ then $$ Df(\vec{x})=\begin{bmatrix}\frac{\partial ...


3

There is the Maurer-Cartan form which is $\mathfrak g$-valued. An example is if we consider the Lie subgroup $G = SO(2) \subset GL(2,\mathbb R) $, we may parametrize $SO(2)$ by $$g(\theta)= \begin{pmatrix}\cos \theta & -\sin\theta \\\sin \theta & \cos \theta\end{pmatrix}\,\, , \theta \in \mathbb R$$ then the matrix of forms (which are functions) is ...


3

If Wikipedia is to be believed, then this is a persymmetric matrix. From the linked article: In mathematics, persymmetric matrix may refer to: 1. a square matrix which is symmetric in the northeast-to-southwest diagonal; or 2. a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.


5

A very common example is the matrix of a plane rotation with angle$\theta$ around the origin: $$\begin{bmatrix}\cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{bmatrix}$$


25

You can define a matrix with elements in any commutative ring, since the only requirement is to be able to perform addition and multiplication with the usual properties. You even may consider the following $2\times 2$ matrices. Such matrices describe the endomorphisms of the direct sum $\;E=U\oplus V$ of two vector spaces $U$ and $V$ $$M=\begin{bmatrix} f_1&...


19

One common use of functions in a matrix is the Hessian matrix in multivariable calculus. This is a matrix of second derivatives with respect to $x_1, x_2, \ldots$. $$ M = \pmatrix{ \frac{\partial^2 f}{\partial^2 x_1} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{...


-2

Using the term "scalar" in the context of Linear Algebra we emphasize that it is an atomic element rather than a collection of elements like "vector" or "matrix". This the main point. In this context we do not care about a nature of such element. Particularly in this context we do not care whether it is a number or not. In this context we do not care if the ...


7

They exist, for example you could have this linear transformation: $L: \mathbb{P}_2\rightarrow M_{2\times 2}, f\rightarrow L(f):=\begin{pmatrix} f'(0) & f(1) \\ \int_{-1}^1f(s)\,ds& 0 \\ \end{pmatrix}$ Functional analysis studies this kind of matrices, I think. Here $\mathbb{P}_2$ represents the set of all polynomial functions from $\mathbb{...


3

There are several reasons I can identify: History. As described by Paul Sinclair, the nomenclature comes from quaternions, where distinguishing between a number and a component of a quaternion is a good idea. This distinction is not so directly relevant to vector spaces, but the name stuck anyhow. Fields other than $\mathbb{R}$ or $\mathbb{C}$. Vector ...


-3

This is probably a poor example, but it would be perfectly valid to have vectors of car models. In that situation, car models would be scalars. Also, "scalar" could be generalized to mean a "scalable data type", which is the requirement for an object to be a valid argument in a linear equation. For instance, all that is needed is multiplication and addition. ...


1

To me, the reason has to do with levels of abstraction. Namely, when we say "number" we instantly carry with it all the baggage of arithmetic and elementary algebra, be it an integer, a rational number or even a complex number. Now, when we say "scalar" we indicate what this thing tends to do with the object (usually, an abstract thing called "vector") ...


0

Let $X$ be a discrete random variable with probability mass function $p_X : \mathcal{X} \to [0,1]$, where $\mathcal{X}$ is a discrete set (possibly countably infinite). Random variable $X$ can be thought of as a continuous random variable with the following probability density function $$f_X (x) = \sum_{x_k \in \mathcal{X}} p_X (x_k) \, \delta (x - x_k)$$ ...


4

Start with etymology: scalar derives from a Latin adjective (scalaris), akin to the word "ladder" (scala). Its first known usage is from François Viète, one of the fathers of "modern algebra". In his In artem analyticem isagoge (Analytic art, 1591), one finds: Magnitudes that ascend or descend proportionally in keeping with their nature from one kind ...


5

You can find this mentioned briefly in the link Barry Cipra provides in a comment on the OP, but the usage of "scalar" in this context comes from quaternions, as does the term "vector". If you are not familiar with quaternions, you can think of them as complex numbers on a bad acid trip. If you are willing to sacrifice commutivity of multiplication, you can ...


15

In mathematics, as well as elsewhere in life, the same thing can be called by different names depending on which aspect of it is of interest to us at the time. For example, I have a roommate who is also a friend; when talking about him I can either call him "my roommate" or "a friend of mine", depending on what is relevant to the conversation. Similarly, ...


0

An interesting read can be found here: When was Matrix Multiplication invented? It is interesting to me that determinants have appeared before matrix algebra or even matrices and that the multiplication rule for determinants predates the discovery of matrix multiplication.


8

Another point is that a vector space can be a field at the same time, for example every field $K$ is trivially a $K$-vector space, and $\mathbb{R}$ is a $\mathbb{Q}$-vector space. Saying number would be ambiguous, so we distinguish between scalars and vectors.


1

Because you multiply the elements of the matrices together. That's the base of the terminology.


44

Not all fields are fields of numbers. For instance, it makes sense to talk about vector spaces over the field of rational functions $\mathbb R(X)$ but the scalars in this case are definitely not numbers.


22

Scalar gives you a sense of what the "number" does. A scalar scales a vector, stretching or contracting each of its coordinates by the same amount. While yes it is a number in common parlance (as long as you are working over a field of numbers, which you probably are), in the context of linear algebra, numbers really just serve this purpose (unless you are ...


11

This is a great question! The point is that we can look at vector spaces over more general objects called fields. The scalars in a vector space are exactly the elements from this field. If the field is the real, rational, or complex numbers, then the scalars are numbers. But we usually don't refer to elements from all fields as numbers.


122

So first of all, "integer" would not be adequate; vector spaces have fields of scalars and the integers are not a field. "Number" would be adequate in the common cases (where the field is $\mathbb{R}$ or $\mathbb{C}$ or some other subfield of $\mathbb{C}$), but even in those cases, "scalar" is better for the following reason. We can identify $c$ in the base ...


2

We don't use integer because not all are integers, I can use $\pi$ as a scalar and it'll be just fine. The reason for scalar vs number is that they are distinct in the sense that a number is an individual entity in a ring while a scalar is an operation on a vector space that changes the vector itself in some sense. In abstract algebra, where linear algebra ...


0

I really feel like I have seen this called "the annihilator of $\mathfrak{a}$ in $M$" and denoted $Ann_M(\mathfrak{a})$, but I have been unable to find a source quickly. That would be a reasonable name in any case.


0

This is easy to show with Hasse diagram. I will first demonstrate a pedagocal example and then this example. Example I use Hasse diagram to describe a poset $(S_2,\geq)$ with a set $$S_2=\{\{d\},\{d,o\},\{d,o,a\},\{d,a\},\{d,a,g\},\{g,o,a,d\}\}$$ where $\{g,o,a,d\}$ is the maximal while $\{d\}$ is the minimal by the containment, here I think they are ...


1

The place this happens naturally is in simple continued fractions. Convergents $p/q$ always have $\gcd(p,q) = 1.$ Given two consecutive convergents, your $a/b$ followed by $c/d,$ the result of the next "digit" being equal to $1$ is precisely that the next convergent is the mediant. If the next "digit" is some $k,$ the new convergent is $$ \frac{a + kc}{b + ...


4

This is called mediant. It appears in statistics, e.g., in Simpson's paradox.


3

The general linear group $GL_n(k)$ is the group of automorphisms (in a suitable sense) of the $n$-space $k^n$. Similarly, the projective linear group $PGL_n(k)$ is the group of automorphisms (in a suitable sense*) of the projective $n$-space $\mathbb{P}^n(k)$. In projective geometry, two nonzero points are identified if they are on a same line going through ...


1

The projective plane over $\mathbb F$ arises from $\mathbb F^3$ by excluding the origin and then identifying vectors that are multiples of each other. Similarly, $PGL_n(\mathbb F)$ arises from the set of $n\times n$ matrices by excluding the singular matrices and then identifying matrices that are multiples of each other (because the center of the general ...


2

It is derived from the Latin word norma, which means rule or standard. ‘Ruler’ — a straight stick that measures length — has the same origin, which is pretty much the same as what a norm does in mathematics.


1

I'd like to add an aspect to your question, that hasn't been touched by any of the other answers thus far: Evidence plays a significant role in current day mathematics. Vectornaut already mentioned one example of this. Let me provide another, that has a different kind of flavour to it. I'd like to talk about large cardinals, but before I do so, let me ...


0

''Mantissa'' is a latin word which means '' something that is added ''. In the context of logarithms it indicate the decimal part of the logarithm of a number, while the ''characteristic'' is the integer part. When I was young there were not home computers ( it seems incredible ?) and the logarithms was calculated using little books called ''logarithm ...


2

"Ordered by containment" means that if we are considering subsets of a set $S$ where $X\subset S$ and $Y\subset S$ then we say that $X\le Y$ (that is, "$X$ is less than $Y$" or "$X$ comes before $Y$") if $X\subset Y$ (that is, $X$ is contained in $Y$, equivalently $X$ is a subset of $Y$). This is not a total order, so there may be no smallest (or minimum) ...


7

If you're a 20th-century sort of scientist, you might describe "evidence" for a hypothesis as a valiant but unsuccessful attempt to falsify it—to prove it wrong by showing that it implies something false. For example, say it's the early 1800s, and you want to test the hypothesis H that Fresnel's wave model accurately describes the behavior of light. You ...


0

I don't know the history of the terminology, but "closed under" does have a relation to the notion of "closed" in topology. The "closed over" terminology is probably a hypercorrection due to wrongly interpreting "under" as a preposition denoting position. We say that some collection $S$ is closed under some $k$-input operation $f$ if the result of applying $...


1

If $f \in C^1$ then it has continuous partial derivatives. Roughly, if $f$ is piecewise $C^1$, then $f:\bigcup X_{i} \rightarrow \bigcup Y_{1}$ is a piecewise function s.t. $f|_{X_{i}} \in C^1.$


5

Evidence is information that tends to indicate that a proposition is more likely to be true, whereas a proof demonstrates conclusively that the proposition is true. As such, yes a proof is evidence but evidence is not necessarily a proof, so you cannot use the terms synonymously. BTW, for this reason the old saw that absence of evidence is not evidence ...


1

In these lecture notes (page 5) the condition seems simply to be that $\phi(A)\in A$ for every $A\in\mathcal F$. As far as I can tell, this is also what Nagata defines -- except that both Nagata and the above notes use a separate directed set $\Delta$ (or $\mathscr D$) instead of $\mathcal F$ itself, but with an ordering given such that it is effectively ...



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