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).

Motivating Warm-Up Problem

• 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.

Historical Milestones in Solving Linear Systems

• $\text{300 BC}$ – 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.

Core Definitions & Concepts

Linear Equation in $n$ Variables

• Standard algebraic form:
  $a<em>1x</em>1 + a<em>2x</em>2 + \dots + a<em>nx</em>n = b$
  where $a_i, b \in \mathbb{R}$ (or $\mathbb{C}$) are known constants.

System of Linear Equations

• A collection of such equations that share the same variables $x<em>1,\dots,x</em>n$.
• 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}

Solution & Solution Set

• A solution is an ordered list $(s<em>1,s</em>2,\dots,s_n)$ that simultaneously satisfies every equation.
• The set of all such ordered lists is the solution set.

Equivalence of Systems

• Two systems are equivalent if they possess exactly the same solution set.

Consistency Classification

• 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.

Matrix Representation

Coefficient Matrix

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

Augmented Matrix

• Appends the constants $b<em>i$ as an extra right-most column: $[A\,|\,\mathbf{b}] = \begin{bmatrix}a</em>{11} &amp; \dots &amp; a<em>{1n} & | & b</em>1\ \vdots &amp; \ddots &amp; \vdots &amp; | &amp; \vdots \ a<em>{m1} & \dots & a</em>{mn} &amp; | &amp; b_m\end{bmatrix}$

Dimension / Size Notation

• “$m \times n$” means m rows and n columns (rows cited first).

Elementary Row Operations (EROs)

1. Replacement (type R1): $R<em>i \leftarrow R</em>i + kR_j$.
2. Interchange (type R2): swap rows $R<em>i$ and $R</em>j$.
3. Scaling (type R3): $R<em>i \leftarrow kR</em>i\,,\; k\neq0$.

• 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.

Fundamental Questions for Any System

• Does at least one solution exist? (Consistency)
• If yes, is that solution unique?

Algorithmic Strategy

• 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.

Example 1 ▸ Unique Solution via Gaussian Elimination

Original (implicit) system labelled (1)–(3) eventually produces the solution $(x<em>1,x</em>2,x_3)=(29,16,3).$ Key steps:

1. Pivot choice: Use $x_1$ in equation (1) as first pivot.
2. Eliminate $x_1$ from equations (2) and (3) using R1 operations.
3. Scale a middle row to obtain a unit pivot for $x_2$.
4. Eliminate $x_2$ from other rows.
5. Achieve triangular (upper-triangular) system.
6. 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}
7. 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.)

Example 2 ▸ Detecting Inconsistency

System labelled (4) is converted by EROs to a triangular augmented matrix that reads, in equation form,
$0x<em>1 + 0x</em>2 + 0x_3 = 1.$

• The statement $0=1$ is a contradiction, therefore no triple $(x<em>1,x</em>2,x_3)$ can satisfy the system.
• Conclusion: the system is inconsistent (solution set = ∅).

Additional Practice Examples (Slide 25)

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

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

Software Aids Mentioned

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

Key Takeaways & Exam Tips

• 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.