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Linear Algebra

Solving a Linear System

A linear algebraic equation is an equation of the system

a1 x1+a2 x2+a3 x3+⋯+an xn=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 s1,s2, and s3that satisfies the equation.

Example

4x1+5x2-2x3=16

is such an equation in which there are three unknown: x1,x2,andx3. One solution to this equation is x1=3,x2=4 and x3=8,since 4*3+5*4-2*8 is equal to 16.

A system of a linear algebraic equation is a set of the equation of the form:

a11 x1+a12 x2+a13 x3+⋯+a1n xn=b1
a21 x1+a22 x2+a23 x3+⋯+a2n xn=b2
a31 x1+a32 x2+a33 x3+⋯+a3n xn=b3
am1 x1+am2 x2+am3 x3+⋯+amn xn=bm

This is called an m*n system of equations; there are m equations and n unknowns.

Matrix Forms

Because 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 right-hand side of the equations:

A           x      =       b
a11 a12 a13...a1n       x1       b1
a21 a22 a23...a2n       x2       b2
a31 a32 a33...a3n       x3       b3
.............................       .............       ........
am1 am2 am3...amn       xn       bm

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:

  • No solutions
  • One solution
  • Infinitely many solutions

Solution using matrix Inverse

Probably 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-1to 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 built-in inv function and matrix multiplication, and also using the "\" operator:

Solving 2x2 Systems of Equations

The 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

Linear Algebra

the determinant D is defined as a11 a22-a12 a21

Linear Algebra

Example

          x1+3x2=-2
          2x1+4x2=1

This would be written in matrix form as

Linear Algebra

The determinant D = 1*4 -3*2 = -2.

Linear Algebra

MATLAB has built-in functions det to find the determinant of the matrix.






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