2From linear systems of equations to matrices

Suppose we have a linear system of equations:

It consists of $m$ equations in the variables  $x_1, \dots, x_n$. The $a_{ij}$ are coefficients in $\R$.

For which values $y_1, \dots, y_m \in \R$ do these equations have a solution $(x_1, \dots, x_n)$? This question certainly depends on the number of equations and their relationship to each other.

Example: The system

does not always have a solution. For instance, it has no solution for $y_1 = 0$ and $y_2 = 1$, since the second equation is two times the first equation. More generally the same reasoning gives that there does not exist a solution whenever $2y_1\neq y_2$. On the other hand it has a solution for $y_1 = 1$ and $y_2=2$ or more generally for $2y_1=y_2$, for example $x_1 = y_1$ and $x_2=0$.

Writing a linear system of equations the way we wrote it above carries a lot of redundancy. For example the variables $x_1, \dots, x_n$ appear in every row. To ease notation, we can use the matrix vector multiplication: We assemble the coefficients $a_{ij}$ into an $(m\times n)$-matrix and write the $x_i$’s and $y_j$’s as vectors.

By definition of matrix vector multiplication, we have $y_i = a_{i1}x_1 + \dots a_{in}x_n$ for each $i \in \{1, \dots, m\}$.

Our example turns into the following matrix vector multiplication