A set of [[homogeneous linear equation]]s can be written as {{tmath|\mathbf A \mathbf x {{=}} \mathbf 0}} for a matrix {{tmath|\mathbf A}}, vector {{tmath|\mathbf x}}, and [[zero vector]] {{tmath|\mathbf x.0}}. A typical situation is that {{tmath|\mathbf A}} is known and a non-zero {{tmath|\mathbf x}} is to be determined which satisfies the equation. Such an {{tmath|\mathbf x}} belongs to {{tmath|\mathbf A}}'s [[Kernel (matrix)|null space]] and is sometimes called a (right) null vector of {{tmath|\mathbf A.}} The vector {{tmath|\mathbf x}} can be characterized as a right-singular vector corresponding to a singular value of {{tmath|\mathbf A}} that is zero. This observation means that if {{tmath|\mathbf A}} is a [[square matrix]] and has no vanishing singular value, the equation has no non-zero {{tmath|\mathbf x}} as a solution. It also means that if there are several vanishing singular values, any linear combination of the corresponding right-singular vectors is a valid solution. Analogously to the definition of a (right) null vector, a non-zero {{tmath|\mathbf x}} satisfying {{tmath|\mathbf x^* \mathbf A {{=}} \mathbf 0}} with {{tmath|\mathbf x^*}} denoting the conjugate transpose of {{tmath|\mathbf x,}} is called a left null vector of {{tmath|\mathbf A.}}
