Let $L:U\rightarrow V$ be a linear mapping, and let $v\in V$ be given. When both the domain $U$ and codomain $V$ are finite-dimensional, a linear equation $$L(u)=v,$$ where $u\in U$ is the unknown, can be solved by means of row reduction. To do so, we need to choose a basis $a_1,\ldots, a_m$ of the domain $U$ , and a basis $b_1,\ldots, b_n$ of the codomain $V$ . Let $M$ be the $n\times m$ transformation matrix of $L$ relative to these bases, and let $y\in\Rset^n$ be the coordinate vector of $v$ relative to the basis of $V$ . Expressing this in terms of matrix notation, we have

Note that the dimension of the domain is the number of variables, while the dimension of the codomain is the number of equations. The equation is called under-determined or over-determined depending on whether the former is greater than the latter, or vice versa. In general, over-determined systems are inconsistent, while under-determined ones have multiple solutions. However, this is a ``rule of thumb'' only, and exceptions are not hard to find. A full understanding of consistency, and multiple solutions relies on the notions of kernel, image, rank, and is described by the rank-nullity theorem.

Remark.