Systems of Linear Equations – Comprehensive Lecture Notes

Administrative & Procedural Announcements

Register on the course e-class platform as soon as possible.
Students who have arranged special transcription services should remain in the lecture hall after class for coordination.
Familiarise yourself with the Socrative polling/quiz system; it will be used during lectures.
A first (trial) online test will be released tomorrow – check the platform.
Begin experimenting with Wolfram Alpha for symbolic/numeric calculations (it will greatly help with homework and demonstrations).

Scenario (expressed in euros, but historically phrased in dollars in many versions):
• Phone + case cost a total of 110.
• Phone alone costs 100 more than the case.
Let p be the phone price and c the case price.
• Equation system:
\begin{cases}
p + c = 110 \
p = c + 100
\end{cases}
• Substitution gives c + 100 + c = 110 \;\Rightarrow\; 2c = 10 \;\Rightarrow\; c = 5.
• Therefore p = 105.
Lesson: even an everyday story translates into two linear equations in two unknowns – the basic object of Linear Algebra.

\text{300 BC} – Babylonians: explicit tables for 2\times2 linear systems.
\text{200 BC} – Chinese “Nine Chapters”: an early elimination technique (precursor of Gaussian elimination).
1545 – Girolamo Cardano (Italy): primitive determinant-like rule later formalised by Cramer.
1683 – Takakazu Seki (Japan) and G. W. Leibniz (Germany): independent conception of determinants.
1750 – Gabriel Cramer (Switzerland): Cramer’s Rule.
1810 – Carl F. Gauss (Germany): systematic row elimination algorithm.
1921 – Emmy Noether (USA/Germany): abstraction to vector spaces, modern axiomatic framework.

Standard algebraic form:
a1x1 + a2x2 + \dots + anxn = b
where a_i, b \in \mathbb{R} (or \mathbb{C}) are known constants.

A collection of such equations that share the same variables x1,\dots,xn.
Example prototype (three equations, three unknowns):
\begin{cases}
a{11}x1 + a{12}x2 + a{13}x3 = b1 \ a{21}x1 + a{22}x2 + a{23}x3 = b2 \
a{31}x1 + a{32}x2 + a{33}x3 = b_3
\end{cases}

A solution is an ordered list (s1,s2,\dots,s_n) that simultaneously satisfies every equation.
The set of all such ordered lists is the solution set.

Exactly one of the following is true for any system:
1. No solution – system is inconsistent.
2. One unique solution – system is consistent.
3. Infinitely many solutions – system is consistent.

Collects only the a{ij} numbers in an m \times n rectangular array. A = \begin{bmatrix}a{11} & a{12} & \dots & a{1n}\ \vdots & \vdots & \ddots & \vdots \ a{m1} & a{m2} & \dots & a_{mn}\end{bmatrix}

Appends the constants bi as an extra right-most column: [A\,|\,\mathbf{b}] = \begin{bmatrix}a{11} & \dots & a{1n} & | & b1\ \vdots & \ddots & \vdots & | & \vdots \ a{m1} & \dots & a{mn} & | & b_m\end{bmatrix}

Properties
• Each operation is invertible.
• Two matrices are row-equivalent if one can be transformed into the other via a finite ERO sequence.
• Row-equivalent augmented matrices represent equivalent linear systems ➜ identical solution sets.

Replace the original system by a row-equivalent one that is easier to interpret:
• Triangular form (forward elimination) for back-substitution.
• Row-Echelon Form (REF) or the stricter Reduced Row-Echelon Form (RREF) for direct reading of solutions.

Original (implicit) system labelled (1)–(3) eventually produces the solution (x1,x2,x_3)=(29,16,3). Key steps:

Pivot choice: Use x_1 in equation (1) as first pivot.
Eliminate x_1 from equations (2) and (3) using R1 operations.
Scale a middle row to obtain a unit pivot for x_2.
Eliminate x_2 from other rows.
Achieve triangular (upper-triangular) system.
Back-substitution:
\begin{aligned}
4x2 &= 12 &\Rightarrow&\; x2 = 3 \
x3 &= 0 \ x1 &= 2x2 + x3 = 6 \Rightarrow x_1 = 29 \text{ (after all numeric updates)}
\end{aligned}
Verification: Substituting (29,16,3) into each original equation returns the stated right-hand sides, confirming correctness.

(Slide snapshots contained arithmetic misalignments; the bullet list captures only the clean logical flow.)

System labelled (4) is converted by EROs to a triangular augmented matrix that reads, in equation form,
0x1 + 0x2 + 0x_3 = 1.

The statement 0=1 is a contradiction, therefore no triple (x1,x2,x_3) can satisfy the system.
Conclusion: the system is inconsistent (solution set = ∅).

The slide lists several 3-variable equations and candidate triples such as (1,0,-1), etc. Suggested exercise routine:

Substitute each candidate triple into every equation.
Confirm whether it satisfies all; if one equation fails, the triple is not a solution.
Attempt EROs to classify each system as in the previous examples.

Socrative ➜ in-class polling, concept checks.
Wolfram Alpha ➜ symbolic row reduction command: RowReduce[{{…}}] or Solve[{…},{x,y,z}].

Every linear modelling story (prices, mixtures, currents, forces) translates to A\mathbf{x}=\mathbf{b}.
Matrices and EROs compress repetitive algebra while guaranteeing equivalence.
Gaussian elimination = algorithmic backbone for existence/uniqueness determination.
Know the three ERO types and be able to reverse any step.
Contradictory rows (e.g., 0\;0\;0\mid c\neq0) signal inconsistency instantly.
A row of all zeros including the augmented part (e.g., 0\;0\;0\mid0) signals a dependent equation ➜ possible infinite solutions if enough free variables remain.
Always perform a quick solution check – arithmetic slips are common under exam pressure.