Linear systems of equations: different perspectives

Click on an arrow to get a description of the connection!

Click on an arrow to get a description of the connection!

Reminder: System of linear equations

Let \(m,n\in\mathbb{N}\). A system of linear equations (LES) in the variables \(x_1,x_2,\ldots,x_n\) is of the form \[\begin{aligned} a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n & = b_1\\ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n & = b_2\\ & \vdots\\ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n & = b_m ~ \end{aligned}\] with \(a_{ij}\) and \(b_i\) being (usually real) numbers. An assignment of values for \(x_1,\ldots,x_n\) such that all equations are satisfied is called a solution of this system of equations. Such a solution is written as a vector.

LES: Looking at rows

Equations describe

• lines (\(\mathbb{R}^2\))

• planes (\(\mathbb{R}^3\))

• hyperplanes (\(\mathbb{R}^n\))

Solution set is their intersection.

LES: Looking at columns

\[\begin{aligned} a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n & = b_1\\ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n & = b_2\\ & \vdots\\ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n & = b_m \end{aligned}\]

\[\Downarrow \text{rearrange}\]

\[\begin{aligned} {\begin{pmatrix} a_{11}\\ a_{21} \\ \vdots \\ a_{m1} \end{pmatrix}} x_1 + { \begin{pmatrix} a_{12}\\ a_{22} \\ \vdots \\ a_{m2} \end{pmatrix}} x_2 + \ldots + { \begin{pmatrix} a_{1n}\\ a_{2n} \\ \vdots \\ a_{mn} \end{pmatrix}} x_n = { \begin{pmatrix} b_{1}\\ b_{2} \\ \vdots \\ b_{m} \end{pmatrix}} \end{aligned}\]