Linear systems of equations

Summary

• A system of \(m\) equations in \(n\) unknowns can be written as

\[a_{1,1}x_1+a_{1,2}x_2+⋯a_{1,n}x_n=b_1\]

\[a_{2,1}x_1+a_{2,2}x_2+⋯a_{2,n}x_n=b_2\]

\[⋯\]

\[a_{m,1}x_1+a_{m,2}x_2+⋯a_{m,n}x_n=b_m\]

• \(a_{1,1},⋯,a_{m,n}\) and \(b_1,⋯,b_m\) are constants while \(x_1,⋯,x_n\) are variables.
• Example:

\[4x_1+2x_2=8\]

\[x_1-2x_2=-3\]

• is system of 2 equations in 2 unknowns, or a 2 by 2 system.
• \(x_1,⋯,x_n\) is called a solution to the system if it solves all equations simultaneously.
• Example: \(x_1=1,x_2=2\) is a solution to the system above.
• A system may have zero, one, or an infinite number of solutions. If it has at least one solution, it is called consistent (otherwise it is called inconsistent).
• A system is called balanced if \(m=n\) , overdetermined if \(m>n\) and underdetermined if \(m<n\) .
• A system called homogeneous if all \(b_1=0,⋯,b_m=0\) .