Two systems of linear equations are equivalent if and only if they have the same set of solutions. In other words, two systems are equivalent if and only if every solution of one of them is also a solution of the other.
All the main methods used to solve linear systems are based on the same principle: given a system, we transform it into an equivalent system that is easier to solve; then, its solution is also the solution of the original system.
Remember that a system of equations in unknowns can be written in matrix form aswhere is the matrix of coefficients of the system, is the vector of constants and is the vector of unknowns.
Definition Let andbe two linear systems, both having an vector of unknowns . The two systems are equivalent if and only if they have the same solutions, that is, if and only if
A system of equations can be transformed into an equivalent one by pre-multiplying both sides of its matrix form by an invertible matrix.
Proposition The system of equations in unknownsis equivalent to the systemfor any invertible matrix .
Suppose solves the systemthat is, and are the same vector. Clearly, if we pre-multiply the same vector by the same matrix , we obtain the same result. As a consequence, which is the same asThus, we have proved thatNow, suppose solves the systemWe have assumed that is invertible, so that its inverse exists. We can pre-multiply both sides of the last equation by and obtainorSince , we haveThus, we have proved thatwhich, together with the implication derived previously, givesthat is, is a solution of one of the two systems if and only if it is a solution of the other one. In other words, the two systems are equivalent.
Example Consider the system of two equations in two unknownsThe system can be written in matrix form aswhereIf we multiply the first equation by and leave the second equation unchanged, we obtain a new systemThe matrix form of the new system iswhereThe new system is equivalent to the original one because the same result can be achieved by pre-multiplying the matrix form of the original system by the invertible matrixIn fact,and
Most of the learning materials found on this website are now available in a traditional textbook format.