A set of m linear equations in n unknowns x₁, x₂,…, x_n that has the form

Here, unlike the inhomogeneous 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 vector x. Thus A is the m × n matrix [a_ij], and

There is always the trivial solution x = 0; in fact, the solutions form a vector space. If m = n, so that A is square then the set of equations has non‐trivial solutions if and only if det A = 0. If m < n, there will necessarily be non-trivial solutions.