## Quick Reference

A set of *m* linear equations in *n* unknowns *x*_{1}, *x*_{2},…, *x*_{n} that has the form

*a*_{11}*x*_{1}+*a*_{12}*x*_{2}+⋯+*a*_{1n}*x*_{n}=0,

*a*_{21}*x*_{1}+*a*_{22}*x*_{2}+⋯+*a*_{2n}*x*_{n}=0,

⋮

*a*_{m1}*x*_{1}+*a*_{m2}*x*_{2}+⋯+*a*_{mn}*x*_{n}=0.

Here, unlike the non-homogeneous case, the numbers on the right-hand sides of the equations are all zero. In matrix notation, this set of equations can be written **Ax**=**0**, where the unknowns form a column matrix **x**. Thus **A** is the *m*×*n* matrix [*a*_{ij}], andIf **x** is a solution of a homogeneous set of linear equations, then so is any scalar multiple *k***x** of it. There is always the trivial solution **x**=**0**. What is generally of concern is whether it has other solutions besides this one. For a homogeneous set consisting of the same number of equations as unknowns, the matrix of coefficients **A** is a square matrix, and the set of equations has non-trivial solutions if and only if det**A**=0.

*Subjects:*
Mathematics.