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The Jacobian matrix represents the differential of f at every point where f is differentiable. In detail, if h is a displacement vector represented by a column matrix, the matrix product J(x) ⋅ h is another displacement vector, that is the best linear approximation of the change of f in a neighborhood of x, if f(x) is differentiable at x.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
Microsoft Math Solver (formerly Microsoft Mathematics and Microsoft Math) is an entry-level educational app that solves math and science problems. Developed and maintained by Microsoft , it is primarily targeted at students as a learning tool.
Some 50 employees joined Amplify. Desmos Studio was spun off as a separate public benefit corporation focused on building calculator products and other math tools. [7] In May 2023, Desmos released a beta for a remade Geometry Tool. In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra ...
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...
The determinant of a matrix A is commonly denoted det(A), det A, or | A |. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism.
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
Other early handheld calculators with symbolic algebra capabilities included the Texas Instruments TI-89 series and TI-92 calculator, and the Casio CFX-9970G. [2] The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a copyleft version of Macsyma is called Maxima. Reduce became free software in ...