Start Date: 10/13/2019
Course Type: Common Course |
Course Link: https://www.coursera.org/learn/linearalgebra2
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Algebra II | Algebra II is generally the first mathematics course that deeply involves abstract and non-linear thinking. |
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. |
Minor (linear algebra) | In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which in turn are useful for computing both the determinant and inverse of square matrices. |
Algebra II | Algebra II (or advanced algebra) is a continuation of Algebra I. The course teaches more fundamentals of algebra and includes logarithms, parabolas, other conic sections, trigonometry, and exponentiation. |
Linear algebra | Functional analysis mixes the methods of linear algebra with those of mathematical analysis and studies various function spaces, such as L spaces. |
Linear algebra | The first four axioms are those of "V" being an abelian group under vector addition. Vector spaces may be diverse in nature, for example, containing functions, polynomials or matrices. Linear algebra is concerned with properties common to all vector spaces. |
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. |
Numerical linear algebra | Numerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, telecommunication, computational finance, materials science simulations, structural biology, data mining, bioinformatics, fluid dynamics, and many other areas. Such software relies heavily on the development, analysis, and implementation of state-of-the-art algorithms for solving various numerical linear algebra problems, in large part because of the role of matrices in finite difference and finite element methods. |
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). |
Nonnegative rank (linear algebra) | In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement that certain coefficients and entries of vectors/matrices have to be nonnegative. |
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. |
Linear algebra | Linear algebra first appeared in American graduate textbooks in the 1940s and in undergraduate textbooks in the 1950s. Following work by the School Mathematics Study Group, U.S. high schools asked 12th grade students to do "matrix algebra, formerly reserved for college" in the 1960s. In France during the 1960s, educators attempted to teach linear algebra through finite-dimensional vector spaces in the first year of secondary school. This was met with a backlash in the 1980s that removed linear algebra from the curriculum. In 1993, the U.S.-based Linear Algebra Curriculum Study Group recommended that undergraduate linear algebra courses be given an application-based "matrix orientation" as opposed to a theoretical orientation. |
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. |
Linear algebra | Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. |
Banach algebra | When the Banach algebra "A" is the algebra L("X") of bounded linear operators on a complex Banach space "X" (e.g., the algebra of square matrices), the notion of the spectrum in "A" coincides with the usual one in the operator theory. For "ƒ" ∈ "C"("X") (with a compact Hausdorff space "X"), one sees that: |
Square (algebra) | The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial is the quadratic polynomial . |
Equation solving | Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra. |