Row View of a System of Linear Equations

Core Idea

2-dimensional

Imagine a system of linear equations in 2-dimensional space.

$$\begin{align} a_{11}x_{1}+a_{12}x_{2}=b_{1} \\ a_{21}x_{1}+a_{22}x_{2}=b_{1} \\ a_{31}x_{1}+a_{32}x_{2}=b_{1} \\ \end{align}$$

Notice that each of the rows is very similar to the familiar equation of a line in a 2-dimensional space ($y=mx+c$). Notice also that the solution for this system of equations is the set of tuples $x_{1}, x_{2}$ that satisfy all the equations simultaneously. Since each of the equations represents a line, the solution for this system of linear equation would be all the points between these “lines” that intersect. They could either (a) not intersect at all, (b) intersect at a single point, or (c) overlap and thus have infinitely many solutions.

3-dimensional

Similarly, imagine a system of linear equations in 3-dimensional space.

$$\begin{align} a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}=b_{1} \\ a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}=b_{2} \\ a_{31}x_{1}+a_{32}x_{2}+a_{33}x_{3}=b_{3} \\ \end{align}$$

Notice that each of these equations represents a plane in 3-dimensional space. Drawing the analogy from the previous example, the solution to these set of linear equations would be a set of points $x_{1}, x_{2}, x_{3}$ that simultaneously lie on each of the planes. This could either be (a) non-existent, (b) meet at a single point, or (c) have infinite values (could intersect on a line or a plane).

Extension

If interpreted row-wise, each of these equations in a system of $n$ unknowns could be imagined as a hyperplane in the n dimensional space. The solution to these equations therefore would be all the points in any of the hyperplanes that also simultaneously lies on every other hyperplane.

Connections

• Forward Links: Column View of a System of Linear Equations, Matrix View of a System of Linear Equations
• Backward Links: System of Linear Equations