Solving Systems Of Linear Equations By Substitution: Fill & Download for Free

There are different methods such asDirect method: which is again sub divided into three auch asElimination MethodSubstitution MethodCross multiplication Method2 .Matrix MethodCrammer’s RuleGauss Elemination MethodGauss-Jordan Elemination MethodTraingularization MethodCholeskey MethodPartition Method3.Iterative methodsJacobi Iterative MethodGauss-Seidel Iterative MethodSOR MethodThese are several methods …and yeh 2 and 3 methods are usually used when problem is big.

Determinants were invented before matrices, so the motivation for defining determinants could not have had anything to do with matrices.Naturally, you're now asking how you could even express a determinant without using a matrix. Florian Cajori's History of Mathematical Notations answers this as it answers many other questions of mathematical notations. From page 87 in volume II, starting with Leibniz:Then Cramer in 1750:and so forth. Bézout, Vandermonde, Laplace, Gauss, Binet, Cauchy, and others also had their notations. A common notation in the 19th century was used by Cauchy and Jacobi:[math]\displaystyle\sum\pm a_{i_1j_1}a_{i_2j_2}\cdots a_{i_nj_n}\tag*{}[/math]which you'll recognize as the permutation definition of determinants, where + or – is chosen depending on whether the permutation indicated in the subscripts was even or odd. In 1841 Cayley used a vertical line notation that became the standard. It's interesting that he found an identity for [math]3\times3[/math] determinants in 1843:So what were these mathematicians doing all this time with determinants before matrices were invented?The first use to solve systems of linear equations. That's what Leibniz and Cramer did. That's now called Cramer's method.Vandermonde had a particular determinant in mind, now called the Vandermonde determinant,Other early applications of determinants were (a) the resultant of two polynomials, a particular determinant, (b) the Jacobian determinant of partial derivatives used for substitution in multivariate calculus, and (c) the Wronskian determinant used in solving systems of linear differential equations.Had the history of mathematics been different and developed along the lines of logic, then matrices (in particular, matrix multiplication) would have been invented before determinants, and then determinants defined for square matrices. That's the way we see it in textbooks. But in mathematics, the more fundamental things often come later. (Indeed, symbolic logic itself wasn't invented until the late 19th century.)

One can describe linear algebra in a few ways. I’ll provide a very concrete answer. It’s meant to be understandable, though it isn’t 100% complete.In algebra, you study equations like [math]6x+5 = 23[/math]. You maybe even study systems of equations, like [math]6x+5y = 23[/math] and [math]6y+5x = 23[/math]. You learn different ways to solve these systems, like graphing them, substitution methods, and possibly others.Linear algebra is a systematic study of systems of equations. You’re not limited to just two or three equations or variables. Potentially, you can have any number.As it turns out, certain mathematical patterns develop. You very quickly realize that matrices and matrix multiplication are important. Moreover, you learn that solving systems of linear equations often just amounts to finding the inverse of a matrix. So you translate a lot of more concrete problems (like “solve this system”) into somewhat more abstract problems (like “invert this matrix.”)It goes on. You learn different ways to manipulate matrices that, in some sense, maintain a certain equivalence. This leads to the abstract idea of a “vector space,” and different ways to represent vectors and/or matrices in vector spaces.But by now, I’m just throwing words at you that probably don’t have a lot of meaning.So suffice it to say, in linear algebra you solve systems of linear equations. In fact, you do that over and over, but in different disguises.