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This approximation is useful because it is easy to compute. But what if I want to do the same thing for a multivariable function? Then I have partial derivatives, but they are only approximation "in one direction" taken separately.

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I you want an approximation of your multivariable function "in all directions at one", you need to find a way to put all of these partial approximation together. That is what formalize linear algebra, and more precisely linear maps between linear vector spaces.

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For instance, with linear algebra, you can deal with spaces of polynomials, of functions That explains the first difficulty in linear algebra: the definition of a vector space is quite abstract at first sight. But it's precisely that abstraction which explains the great power of linear algebra and it's often the case in maths in general because it's that abstraction which affords applying the same language and the same theorems to so different mathematical objects.

It's quite frequent to hear, at a higher level, that something "is simply linear algebra" to say that what remains to do should be easy for every mathematician listening to this conversation. Moreover, and simpler than the other points I mentioned: we need more than 2 or 3 dimensions! Many examples:.


It is mostly about starting with the study of solutions of systems of linear equations and continues with the development of matrix notation up to the axiomatic definition of vector spaces. Another line is analytic geometry, or vector geometry, combining algebra and geometry. See for example:.

Nicolas Bourbaki went a step further: he omitted geometric terms like point, line, etc. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Why we have introduced linear algebra?

Linear algebra

Asked 1 year, 1 month ago. Active 1 year, 1 month ago. Viewed 1k times. For instance, your favourite websites like Netflix rely on linear algebra matrices and operations with them and so on. The fundamental strategy of calculus is to approximate a nonlinear function locally using a linear transformation. But understanding the linear transformation requires linear algebra.

Linear Algebra 13e: The LU Decomposition

The geometrical aspects of linear algebra will be maybe, I don't know One should not have to read through a lot of comments to find out what distinguishes your question from the other one. You could edit further. Pece Pece 8, 1 1 gold badge 13 13 silver badges 41 41 bronze badges.

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The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple but non-trivial theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.

This course is designed to prepare students to solve linear algebra problems arising from many applications such as mathematical models of physical or engineering processes. Students are introduced to modern concepts and methodologies in numerical linear algebra, with particular emphasis on the methods that can be used to solve very large-scale problems. In-depth discussion of theoretical aspects such as stability and convergence will be used to enhance student understanding of the numerical methods.

Students will also be required to perform some programming and computation so as to gain experience in implementing and observing the numerical performance of the various numerical methods. Numerical matrix algebra, orthogonalization and least squares methods, unsymmetric and symmetric eigenvalue problems, iterative methods, advanced solvers for partial differential equations.