Start Date: 07/05/2020
Course Type: Common Course |
Course Link: https://www.coursera.org/learn/linearalgebra2
Doğrusal cebir ikili dizinin ikincisi olan bu ders birinci derste verilen temel bilgilerin üzerine eklemeler yapılarak tamamen matris işlemleri ve uygulamalarını kapsamaktadır. Cebirsel denklem sistemleri, sonuçların tekilliği ve var olup olmadığı, determinantlar ve onların doğal olarak nasıl oluştuğu, öz değer problemleri ve onların matris fonksiyonlarına uygulanışı vb. konulara derste değinilmektedir. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır. Bölümler: Bölüm 1: Doğrusal Cebir I'in Özeti Bölüm 2: Kare Matrislerde Determinant Bölüm 3: Kare Matrislerin Tersi Bölüm 4: Kare Matrislerde Özdeğer Sorunu Bölüm 5: Matrislerin Köşegenleştirilmesi Bölüm 6: Matris Fonksiyonları Bölüm 7: Matrislerle Diferansiyel Denklem Takımları ----------- This second of the sequence of two courses builds on the fundamentals of the first course, is entirely on matrix algebra and applications. Specifically, the studies include systems of algebraic equations including the existence and uniqueness of solutions, determinants and how they arise naturally, eigenvalue problems with their applications to diagonalization and matrix functions. The course is designed in the same spirit as the first one with a “content based” emphasis, answering the “why” and “where“ of the topics, as much as the traditional “what” and “how” leading to “definitions” and “proofs”. Chapters: Chapter 1: Summary of Linear Algebra I Chapter 2: Determinant Chapter 3: Inverse of Square Matrices Chapter 4: Eigenvalue Problem in Square Matrices Chapter 5: Diagonalization of Matrices Chapter 6: Matrix Functions Chapter 7: Matrices and Systems of Differential Equations ----------- Kaynak: Attila Aşkar, “Doğrusal cebir”. Bu kitap dört ciltlik dizinin üçüncü cildidir. Dizinin diğer kitapları Cilt 1 “Tek değişkenli fonksiyonlarda türev ve entegral”, Cilt 2: "Çok değişkenli fonksiyonlarda türev ve entegral" ve Cilt 4: “Diferansiyel denklemler” dir. Source: Attila Aşkar, Linear Algebra, Volume 3 of the set of Vol1: Calculus of Single Variable Functions, Volume 2: Calculus of Multivariable Functions and Volume 4: Differential Equations.
Doğrusal cebir ikili dizinin ikincisi olan bu ders birinci derste verilen temel bilgilerin üzerine e
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Trace (linear algebra) | The kernel of this map, a matrix whose trace is zero, is often said to be ' or ', and these matrices form the simple Lie algebra "sl", which is the Lie algebra of the special linear group of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the special linear Lie algebra is the matrices which "infinitesimally" do not change volume. |
Linear algebra | In 1882, Hüseyin Tevfik Pasha wrote the book titled "Linear Algebra". The first modern and more precise definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The use of matrices in quantum mechanics, special relativity, and statistics helped spread the subject of linear algebra beyond pure mathematics. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. |
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Linear algebra | Functional analysis mixes the methods of linear algebra with those of mathematical analysis and studies various function spaces, such as L spaces. |
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Special linear Lie algebra | In mathematics, the special linear Lie algebra of order n (denoted formula_1) is the Lie algebra of formula_2 matrices with trace zero and with the Lie bracket formula_3. This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. The Lie group that it generates is the special linear group. |
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Linear algebra | If, in addition to vector addition and scalar multiplication, there is a bilinear vector product , the vector space is called an algebra; for instance, associative algebras are algebras with an associate vector product (like the algebra of square matrices, or the algebra of polynomials). |
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Pauli matrices | The group SU(2) is the Lie group of unitary 2×2 matrices with unit determinant; its Lie algebra is the set of all 2×2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra formula_42 is the 3-dimensional real algebra spanned by the set }. In compact notation, |
Trace (linear algebra) | Conversely, any square matrix with zero trace is a linear combinations of the commutators of pairs of matrices. Moreover, any square matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros. |
Trace (linear algebra) | characterize the trace completely in the sense that follows. Let "f" be a linear functional on the space of square matrices satisfying . Then "f" and tr are proportional. |
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Linear algebra | The set of points with coordinates that satisfy a linear equation forms a hyperplane in an "n"-dimensional space. The conditions under which a set of "n" hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. |
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Equation solving | Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra. |