*Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties.*

Linear algebra is concerned with properties common to all vector spaces.

When a bijective linear map exists between two vector spaces (that is, every vector from the second space is associated with exactly one in the first), the two spaces are isomorphic.

Until the 19th century, linear algebra was introduced through systems of linear equations and matrices.

In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract.

All these questions can be solved by using Gaussian elimination or some variant of this algorithm.

## Solved Problems In Linear Algebra

The study of subsets of vector spaces that are themselves vector spaces for the induced operations is fundamental, similarly as for many mathematical structures. More precisely, a linear subspace of a vector space is a vector space.) For example, the image of a linear map, and the inverse image of 0 by a linear map (called kernel or null space) are linear subspaces.In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule.Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy.Its use is illustrated in eighteen problems, with two to five equations.Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry.Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions.Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models.For nonlinear systems, which cannot be modeled with linear algebra, linear algebra is often used as a first-order approximation.The procedure for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art.Linear algebra grew with ideas noted in the complex plane.For instance, two numbers w and z in ℂ have a difference w – z, and the line segments are of the same length and direction. The four-dimensional system ℍ of quaternions was started in 1843.

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