User:Phlsph7/Algebra - Linear algebra

Linear algebra

Main article: Linear algebra

Linear algebra employs the methods of elementary algebra to study systems of linear equations. An equation is linear if no variable is multiplied with another variable and no operations like exponentiation, extraction of roots, and logarithm are applied to variables. For example, the equations $0.25x-4=y$ and $x_{1}-7x_{2}+3x_{3}=0$ are linear while the equations $x^{2}=y$ and $3x_{1}x_{2}+15=0$ are non-linear. Several equations form a system of equations if they all rely on the same set of variables.

Systems of linear equations are often expressed through matrices and vectors to represent the whole system in a single equation. This can be done by moving the variables to the left side of each equation and moving the constant terms to the right side. The system is then expressed by formulating a matrix that contains all the coefficients of the equations and multiplying it with the vector made up of the variables. For example, the system of equations

1. $9x_{1}+3x_{2}-13x_{3}=0$
2. $2.3x_{1}+7x_{3}=9$
3. $-5x_{1}-17x_{2}=-3$

can be written as

$${\begin{bmatrix}9&3&-13\\2.3&0&9\\-5&-17&0\end{bmatrix}}\cdot {\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}0\\9\\-3\end{bmatrix}}$$

Like elementary algebra, linear algebra is interested in manipulating and transforming equations to solve them. It goes beyond elementary algebra by dealing with several equations at once and looking for the values for which all equations are true at the same time. For example, if the system is made of the two equations $3x_{1}-x_{2}=0$ and $x_{1}+x_{2}=8$ then using the values 1 and 3 for $x_{1}$ and $x_{2}$ does not solve the system of equations because it only solves the first but not the second equation.

Two central questions in linear algebra are whether a system of equations has any solutions and, if so, whether it has a unique solution. A system of equations that has solutions is called consistent. This is the case if the equations do not contradict each other. If two or more equations contradict each other, the system of equations is inconsistent and has no solutions. For example, the equations $x_{1}-3x_{2}=0$ and $x_{1}-3x_{2}=7$ contradict each other since no values of $x_{1}$ and $x_{2}$ exist that solve both equations at the same time.

Whether a consistent system of equations has a unique solution depends on the number of variables and the number of independent equations. Several equations are independent of each other if they do not provide the same information and cannot be derived from each other. A unique solution exists if the number of variables is the same as the number of independent equations. Underdetermined systems, by contrast, have more variables than equations and have an infinite number of solutions if they are consistent.

Linear equations with two variables can be interpreted geometrically as lines. The solution of a system of linear equations is where the lines intersect.

Many of the techniques employed in elementary algebra to solve equations are also applied in linear algebra. The substitution method starts with one equation and isolates one variable in it. It proceeds to the next equation and replaces the isolated variable with the found expression, thereby reducing the number of unknown variables by one. It applies the same process again to this and the remaining equations until the values of all variables are determined. The elimination method creates a new equation by adding one equation to another equation. This way, it is possible to eliminate one variable that appears in both equations. For a system that contains the equations $-x+7y=3$ and $2x-7y=10$, it is possible to eliminate y by adding the first to the second equation, thereby revealing that x is 13. Many advanced techniques implement algorithms based on matrix calculations, such as Cramer's rule, the Gauss–Jordan elimination, and LU Decomposition.

On a geometric level, systems of equations can be interpreted as geometric figures. For systems that have two variables, each equation represents a line in two-dimensional space. The point where the two lines intersect is the solution. For inconsistent systems, the two lines run parallel, meaning that there is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to graphically look for solutions by plotting the equations and determining where they intersect. The same principles also apply to systems of equations with more variables, with the difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to planes in three-dimensional space and the points where all planes intersect solve the system of equations.

References

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