Definition of system of linear algebraic equations, the augmented and matrix forms of its entry, a solution, consistent or inconsistent, determined or undetermined system

Lecture plan

Lecture 2. Systems of linear algebraic equations

2.1. Definition of system of linear algebraic equations, the augmented and matrix forms of its entry, a solution, consistent or inconsistent, determined or undetermined system

2.2. Solution of system of equations using Cramer method

2.3. Solution of system of equations using an inverse matrix

2.4. Solution of system of equations using Jordan–Gauss method

2.5. Investigation of the system compatibility with the help of. Kronecker-Capelli theorem

A system of linear algebraic equations with unknowns has the form:

, (2.1)

where are the coefficients of the system; are its free terms and are the unknowns.

A solution of the system (2.1) is a set of numbers satisfying every equation of system.

Every system of the form (2.1) has either no solution, one solution or infinitely many solutions.

A system is consistent or compatible if there exists at least one solution, otherwise it is inconsistent or incompatible.

A compatible system is called definite or determined if it has the only solution.

A compatible system is called indefinite or undetermined if it has more than one various solutions.

Systems are equivalent if they have the same solution set.

If for all j the system (2.1) is homogeneous.

Note that the system (2.1) can be written in a matrix form as , where the -matrix is the matrix of coefficients or the basic matrix, the matrix-column is a matrix-column of free terms and the matrix-column is the unknown matrix-column.

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