Algebra Notes 2

2.1) Systems of linear equations

$a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1$

$a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2$

$\dots$

$a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n = b_m$

This is a system of m linear equations with n unknowns. 

a) The matrix of this system is defined as follows

$$A = \begin{bmatrix}a_{11}&a_{12}&...&a_{1n}\\a_{21}&a_{22}&...&a_{2n}\\...&...&...&...\\a_{m1}&a_{m2}&...&a_{mn}\end{bmatrix}$$

На български: матрица на системата

b) The augmented matrix of this system is defined as follows

$$B = \begin{bmatrix}a_{11}&a_{12}&...&a_{1n}&b_1\\a_{21}&a_{22}&...&a_{2n}&b_2\\...&...&...&...&...\\a_{m1}&a_{m2}&...&a_{mn}&b_m\end{bmatrix}$$

На български: разширена матрица на системата

2.2) Types of systems of linear equations

a) Independent system: has exactly one solution

b) Inconsistent system: has no solutions

c) Consistent system: has at least one solution

d) Dependent system: has infinitely many solutions

a) Определена система: има точно едно решение

b) Несъвместима система: няма решения

c) Съвместима систeмa: има поне едно решение

d) Неопределена система: има безбройно много решения

2.3) Elementary transformations applied to a system of linear equations

a) swapping two rows

b) multiplying a row of the system with a non-zero number

$R := \lambda R$, where $\lambda \neq 0$ 

c) adding to a row another row (multiplied by a number)

$R_2 := R_2 + \lambda R_1$

Note: If we apply (to a given system of linear equations) a finite number of elementary transformations, the resulting system is equivalent to the original system.

2.4) Gaussian elimination - a general method for solving systems of linear equations

Gaussian elimination