Linear AlgebraSolving a Linear SystemA linear algebraic equation is an equation of the system a_{1} x_{1}+a_{2} x_{2}+a_{3} x_{3}+⋯+a_{n} x_{n}=b where a's are constant coefficients, the x's are the unknowns, and b is a constant. A solution is a sequence of numbers s_{1},s_{2}, and s_{3}that satisfies the equation. Example 4x_{1}+5x_{2}2x_{3}=16 is such an equation in which there are three unknown: x_{1},x_{2},andx_{3}. One solution to this equation is x_{1}=3,x_{2}=4 and x_{3}=8,since 4*3+5*42*8 is equal to 16. A system of a linear algebraic equation is a set of the equation of the form: a_{11} x_{1}+a_{12} x_{2}+a_{13} x_{3}+⋯+a_{1n} x_{n}=b_{1} This is called an m*n system of equations; there are m equations and n unknowns. Matrix FormsBecause of the method that matrix multiplication works, these equations can be defined in the matrix form as Ax = b where A is the matrix of the coefficients, x is the column vector of the unknown, and b is a column vector of the constant from the righthand side of the equations: A x = b A solution set is a set of all possible solutions to the system of equation (all sets of value for the unknowns that solve the equation). All systems of linear equations have either:
Solution using matrix InverseProbably the simple way of solving this system of equations is to use the matrix inverse. A^{1} A=1 We can multiply both sides of the matrix equation AX= B by A^{1}to get A^{1} AX=A^{1} B Or X=A^{1} B So, the solution can be found as a product of the inverse of A and the column vector b. In MATLAB, there are two method of doing this, using the builtin inv function and matrix multiplication, and also using the "\" operator: Solving 2x2 Systems of EquationsThe simplest system is a 2 x 2 system, with just two equations and two unknowns. For these systems, there is a simple definition for the inverse of a matrix, which uses the determinant D of the matrix. For a coefficient matrix, A generally defined as the determinant D is defined as a_{11} a_{22}a_{12} a_{21} Example x_{1}+3x_{2}=2 This would be written in matrix form as The determinant D = 1*4 3*2 = 2. MATLAB has builtin functions det to find the determinant of the matrix.
Next TopicGauss and GaussJordan Elimination
