Gist – Page 4 – Ruficalix

What is the Gist of a “Manifold”?

“In mathematics, a manifold is a topological space that locally resembles Euclidean space* near each point” (“Manifold,” Wikipedia, retrieved 6/10/2020). In other words, if you zoom in close enough to any point on a manifold, you won’t be able to tell the difference between what you see and a Euclidean space*. For example, if you zoom in close enough to a circle, it will look almost exactly like a straight line (ibid). This also works for most of a figure 8, but if you focus on the middle of the figure 8, no amount of zooming in will ever make it look like a single straight line (ibid).

There exists something called “Manifolds with a boundary.” “A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an [n-dimensional ‘manifold with a boundary’] is an (n−1)-[dimensional] manifold… A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold (ibid, italics added).”

*What is the Gist of “Euclidean Space”?

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of “Euclidean Space”?

A 3-dimensional Euclidean space (which is normally what is being referred to if talking about a Euclidean space) is basically the space in which we live (excluding relativistic effects), where moving and rotating things don’t change them and you can use Cartesian coordinates to say where things are. The 2-dimensional Euclidean space is basically a plane, and a 1-dimensional Euclidean space is basically a straight line. Other dimensions of Euclidean space are allowed.

Source: “Euclidean space,” Wikipedia, retrieved 6/10/2020

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Vector Space”?

(The first sentence is the most important in my opinion)

“A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied (“scaled”) by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field*. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms [(listed in § Definition)]. For specifying that the scalars are real or complex numbers, the terms real vector space and complex vector space are often used.”

“A vector space over a field* F is a set V together with two [certain] operations…

“The first operation, called vector addition or simply addition… takes any two vectors v and w [in V]and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (The resultant vector is also an element of the set V.)
“The second operation, called scalar multiplication… takes any scalar a [in F] and any vector v [in V] and gives another vector av. (Similarly, the vector av is an element of the set V. Scalar multiplication is not to be confused with the scalar product, also called inner product or dot product, which is an additional structure present on some specific, but not all vector spaces. Scalar multiplication is a multiplication of a vector by a scalar; the other is a multiplication of two vectors producing a scalar.)”

Quotes taken from “Vector space,” Wikipedia, retrieved 6/10/2020

*What is the Gist of a “Field” (in mathematics)?

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Field” (in mathematics)?

“In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do…

“The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers.” (“Field (mathematics),” Wikipedia, retrieved 6/10/2020) There are many other fields, but they are too complicated for this post. See that Wikipedia article for some of them.

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of the “Conjugate Transpose” (of a Matrix)?

“In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix ${\boldsymbol {A}}$ with complex entries is the n-by-m matrix ${\boldsymbol {A}}^{\mathrm {H} }$ obtained from ${\boldsymbol {A}}$ by taking the transpose [of A] and then taking the complex conjugate of each entry [in that transpose]. (The complex conjugate of $a+ib$ , where $a$ and $b$ are real numbers, is $a-ib$ .)” (“Conjugate transpose,” Wikipedia, retrieved 6/9/2020)

“The reason we want to do this [as opposed to just taking the conjugate] is so that we can multiply the matrix and the conjugate transpose. Simply taking the conjugate will not give us matrices we can multiply if they are not square.” (“Complex, Hermitian, and Unitary Matrices,” Professor Dave Explains)

In addition, it seems like the conjugate transpose is fundamental in defining several other matrices (as talked about here) and has several useful properties (such as these)

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of “Quadratic Form”?

“In mathematics, a quadratic form is a polynomial with terms all of degree two. For example, $4x^2 + 2xy - 3y^2$ .” (“Quadratic form“, Wikipedia, retrieved 6/8/2020)

“Quadratic forms are not to be confused with a quadratic equation which has only one variable and includes terms of degree two or less”(ibid)

Also, this video from Khan Academy was helpful: Expressing a quadratic form with a matrix

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Matrix-Valued Function”?

A matrix-valued function is a matrix which, instead of having simple numbers in each component, has functions. Here’s another way to think of it: putting independent variables into this function will lead to you getting a matrix out of the function.

Source: “5.1 Matrix-valued functions” by DarrenOngCL

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of “Quadratic Approximation”?

Some functions are easier to work with than others, especially when using computers. Of the different types of functions, polynomials are some of the easiest to work with. Quadratic approximation is a way of approximating a more complicated function as a second-order polynomial. In addition to making sure the first derivative of the approximation at a point is equal to the first derivative of the actual function at that point (which is what you do in linear approximations), quadratic approximations make sure that the second derivative of the approximation at that point matches the second derivative of the function at that point. This leads to a better approximation than simply using linear approximation. Note that the accuracy of the approximation generally goes down the farther away you get from the point where you based the function.

Sources (all Khan Academy):

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of “Ill-Posed Problems?”

“An ill posed problem is one which doesn’t meet the three Hadamard criteria for being well-posed. These criteria are:” (“Ill Posed Problem: Definition,” Statistics How To, retrieved 6/4/2020)

“a solution exists,
“the solution is unique,
“the solution’s behaviour changes continuously with the initial conditions*” (“Well-posed problem,” Wikipedia, retrieved 6/4/2020)

If the problem does match all three of those criteria, it is considered to be a “well-posed” problem. (ibid.)

“Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers.” (ibid.)

Other sources:

“ill-posed problem,” The Free Dictionary, retrieved 6/4/2020
“Ill-posed problems,” Encyclopedia of Mathematics, retrieved 6/4/2020

*Small changes to the inputs of the problem result in (relatively) small changes to the output of the problem no matter what inputs you are using. This doesn’t happen if a small change results in a dramatic output, such as with a step function that has an output of 0 if the input is less than 1 and an output of 100 if the input is greater than or equal to 1.

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Metric Space” or a “Metric” (Mathematics)?

“A metric space is an ordered pair $(M,d)$ where $M$ is a set and $d$ is a metric on $M$ .”

“The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

“the distance from a point to itself is zero,
“the distance between two distinct points is positive,
“the distance from A to B is the same as the distance from B to A, and
“the distance from A to B (directly) is less than or equal to the distance from A to B via any third point C.”

“The function $d$ is also called distance function or simply distance. Often, $d$ is omitted [when describing the metric space symbolically,] and one just writes $M$ … if it is clear from the context what metric is used.”

(Quotes selected from “Metric space,” Wikipedia, retrieved 6/4/2020)

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).