The course begins with two concrete algebraic systems — complex numbers and vectors in $\mathbb{R}^3$ — that arise independently in algebra and geometry. Complex numbers extend the real line to an algebraically closed field where every polynomial has roots; vectors in three dimensions provide the language of forces, velocities, and electromagnetic fields. The first surprise of the course is that these apparently unrelated systems share a common algebraic structure: both support addition and scalar multiplication satisfying the same axioms. Abstracting this structure gives the notion of a *vector space*, and with it comes a coordinate-free theory of dimension, bases, and linear independence that applies uniformly to $\mathbb{R}^n$, polynomial spaces, function spaces, and matrix spaces.

The second half of the course studies *[linear maps](/page/Linear%20Map)* — the structure-preserving transformations between vector spaces — and their matrix representations. The central question becomes: given a linear map, can we choose coordinates in which it takes the simplest possible form? This is the diagonalisation problem, and its solution requires the theory of eigenvalues, determinants, and the characteristic polynomial. The course culminates in the spectral theorem for real symmetric matrices, which guarantees not just diagonalisation but an *orthonormal* eigenbasis, with applications to quadratic forms, coupled differential equations, and the classification of conics.

Prerequisites are limited to fluency with real numbers, basic set notation, and elementary algebra. No prior knowledge of complex numbers, abstract algebra, or analysis is assumed.

# Complex Numbers

The real number line, for all its power, has a fundamental algebraic gap: not every polynomial equation has a solution. The equation $x^2 + 1 = 0$ demands a number whose square is $-1$, and no real number has this property — the square of any real number is non-negative. This is not merely an inconvenience. Without roots of $x^2 + 1 = 0$, the factorisation $x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x-1)(x+1)(x^2+1)$ terminates prematurely: four roots are expected from a degree-four polynomial, but only two are visible. To restore the algebraic completeness that the real numbers lack, we adjoin a single new element $i$ satisfying $i^2 = -1$ and extend the field operations accordingly.

## The Complex Field

The key observation is that if we want $i^2 = -1$ and we want the usual laws of arithmetic to hold, then every expression built from real numbers and $i$ simplifies to the form $x + iy$ with $x, y \in \mathbb{R}$, since higher powers of $i$ cycle: $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and so on. This motivates the following definition.

[definition:Complex Number] A **complex number** is an ordered pair $(x, y) \in \mathbb{R}^2$, written $z = x + iy$, where $i$ is a formal symbol satisfying $i^2 = -1$. The real number $x = \operatorname{Re}(z)$ is the **real part** and $y = \operatorname{Im}(z)$ is the **imaginary part**. The [set](/page/Set) of all complex numbers is denoted $\mathbb{C}$. The real numbers $\mathbb{R}$ embed in $\mathbb{C}$ via the identification $x \mapsto x + i \cdot 0$. [/definition]

Note that the imaginary part of $z = 3 + 2i$ is $2$, not $2i$ — the imaginary part is a real number.

The arithmetic operations on $\mathbb{C}$ are forced by the requirement that the usual distributive, commutative, and associative laws hold together with $i^2 = -1$.

[definition:Complex Addition] For $z = x + iy$ and $w = u + iv$ in $\mathbb{C}$, the **sum** is \begin{align*} z + w = (x + u) + i(y + v). \end{align*} [/definition]

[definition:Complex Multiplication] For $z = x + iy$ and $w = u + iv$ in $\mathbb{C}$, the **product** is \begin{align*} zw = (xu - yv) + i(xv + yu). \end{align*} [/definition]

This multiplication rule is not arbitrary — it is the unique formula obtained by expanding $(x + iy)(u + iv) = xu + ixv + iyu + i^2yv$ and using $i^2 = -1$. With these operations, $\mathbb{C}$ forms a field: every nonzero complex number has a multiplicative inverse.

How do we compute $1/z$? The naive approach of dividing by $x + iy$ is meaningless, but there is a trick: multiply numerator and denominator by the **conjugate** of the denominator.

[definition:Complex Conjugate] For $z = x + iy \in \mathbb{C}$, the **complex conjugate** is $\overline{z} = x - iy$. [/definition]

The conjugate reverses the sign of the imaginary part while preserving the real part. Its key algebraic property is that $z\overline{z} = x^2 + y^2$ is always a non-negative real number: the imaginary cross-terms cancel. This immediately yields the formula for division:

\begin{align*} \frac{1}{z} = \frac{\overline{z}}{z\overline{z}} = \frac{x - iy}{x^2 + y^2}, \end{align*}

valid whenever $z \neq 0$ (i.e., not both $x = 0$ and $y = 0$). The quantity $x^2 + y^2$ appearing in the denominator deserves its own name.

[definition:Modulus] For $z = x + iy \in \mathbb{C}$, the **modulus** is $|z| = \sqrt{x^2 + y^2} = \sqrt{z\overline{z}}$. [/definition]