Theory

Linear systems and rank

How to write a linear system in matrix form, how to compute the rank, when a system has solutions and how many, how to solve it with Gauss or with Cramer, and how to discuss it as a parameter varies.

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Linear systems and matrix form
Geometric interpretation

Linear systems and matrix form

1. Linear system

A linear system of $m$ equations in $n$ unknowns $x_{1}, \dots, x_{n}$ is a set of equations to be satisfied simultaneously

⎩ ⎨ ⎧ a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} = b_{1} a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} = b_{2} ⋮ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{mn} x_{n} = b_{m}

with coefficients $a_{ij} \in R$ and constant terms $b_{i} \in R$ . A solution is an $n$ -tuple $(x_{1}, \dots, x_{n}) \in R^{n}$ that satisfies all the $m$ equations, and solving the system means determining the set of all solutions. The equations are linear because the unknowns appear only to the first degree and are not multiplied together.

2. Consistent and inconsistent system

A linear system is called

consistent if it has at least one solution, inconsistent if it has none;
determined if it is consistent with exactly one solution, underdetermined if it is consistent with more than one solution.

The system formed by $x + y = 1$ and $x + y = 3$ is inconsistent, because the same quantity $x + y$ cannot equal both $1$ and $3$ . An underdetermined system always has infinitely many solutions, as follows from the Rouché-Capelli theorem; consequently a linear system with exactly two solutions does not exist.

3. Matrix form

The system is written in compact form as

A x = b, A = (a_{ij}) \in M_{m, n} (R), x = x_{1} ⋮ x_{n}, b = b_{1} ⋮ b_{m},

where $A$ is the coefficient matrix and the product is the row-by-column product, for which see the topic on matrices. The augmented matrix $(A ∣ b) \in M_{m, n + 1} (R)$ is obtained by appending the column of constant terms to $A$ . Row $i$ of the augmented matrix contains all the information of equation $i$ , therefore every operation on the equations translates into an operation on the rows of $(A ∣ b)$ .

Example

The system

{x + 2 y - z = 3 2 x - y + z = 1

has $m = 2$ equations and $n = 3$ unknowns, with

A = (12 2 - 1 - 1 1), (A ∣ b) = (12 2 - 1 - 1 1 31) .

Geometric interpretation

An equation $a x + b y = c$ with $(a, b) \neq = (0, 0)$ represents a line in the plane, so solving a system of two equations in two unknowns means finding the points common to two lines. There are three possible cases.

The lines are intersecting: the system is determined and the solution is the point of intersection. This happens when the coefficients of the unknowns are not proportional, that is, when $det A \neq = 0$ .
The lines are parallel and distinct: the system is inconsistent.
The lines are coincident: the two equations are proportional and the system has infinitely many solutions, the points of the line.

y = 3 - x

y = x - 1

The system formed by

x + y = 3

and

x - y = 1

is determined: the two lines meet only at the point

(2, 1)

y = 3 - x

y = 1 - x

The system formed by

x + y = 3

and

x + y = 1

is inconsistent: the two lines are parallel and distinct

In three unknowns each equation represents a plane in space, and three planes can have in common a point, a line, a plane, or no point at all; for the study of their relative positions, see the topic on lines and planes in space.

The theory continues with 7 more sections (elementary operations and gaussian elimination, the rank of a matrix, the bordered minors theorem…), plus all worked examples, the solutions and the PDF.

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