Tag: Math

(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

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

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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 :).

“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.

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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 :).

“In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix  with complex entries is the n-by-m matrix  obtained from  by taking the transpose [of A] and then taking the complex conjugate of each entry [in that transpose]. (The complex conjugate of , where  and  are real numbers, is .)” (“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)

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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 :).