2From linear systems of equations to matrices

Suppose we have a linear system of equations:

It consists of $mm$ equations in the variables  $x_1,\dots ,x_{n}x\_1,\; \backslash dots,\; x\_n$. The $a_{ij}a\_\{ij\}$ are coefficients in $\mathbb{R}\backslash R$.

For which values $y_1,\dots ,y_m\in \mathbb{R}y\_1,\; \backslash dots,\; y\_m\; \backslash in\; \backslash R$ do these equations have a solution $(x_1,\dots ,x_n)(x\_1,\; \backslash dots,\; x\_n)$? 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 $y_1=0y\_1\; =\; 0$ and $y_2=1y\_2\; =\; 1$, since the second equation is two times the first equation. More generally the same reasoning gives that there does not exist a solution whenever $2y_1\mathrm{\ne }{y}_{2}2y\_1\backslash neq\; y\_2$. On the other hand it has a solution for $y_1=1y\_1\; =\; 1$ and $y_2=2y\_2=2$ or more generally for $2y_1=y_{2}2y\_1=y\_2$, for example $x_1=y_{1}x\_1\; =\; y\_1$ and $x_2=0x\_2=0$.

Writing a linear system of equations the way we wrote it above carries a lot of redundancy. For example the variables $x_1,\dots ,x_{n}x\_1,\; \backslash dots,\; x\_n$ appear in every row. To ease notation, we can use the matrix vector multiplication: We assemble the coefficients $a_{ij}a\_\{ij\}$ into an $(m\times n)(m\backslash times\; n)$-matrix and write the $x_{i}x\_i$’s and $y_{j}y\_j$’s as vectors.

By definition of matrix vector multiplication, we have $y_i=a_{i1}{x}_1+\dots a_{in}{x}_{n}y\_i\; =\; a\_\{i1\}x\_1\; +\; \backslash dots\; a\_\{in\}x\_n$ for each $i\in \{1,\dots ,m\}i\; \backslash in\; \backslash \{1,\; \backslash dots,\; m\backslash \}$.

Our example turns into the following matrix vector multiplication