First third

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+#set document(
+  title: [Lecture 1],
+  author: "Gidon Rosalki",
+  date: auto,
+)
+#set heading(numbering: "1.")
+#set text(font: "New Computer Modern")
+#set page(margin: (
+  bottom: 0.5cm,
+  right: 1.5cm,
+  left: 1.5cm,
+  top: 1.5cm,
+))
+#set list(indent: 10pt)
+#set enum(indent: 10pt)
+
+#show title: set align(center)
+#show link: set text(rgb("005eff"))
+#show link: underline
+#show math.equation.where(block: true): set block(breakable: true)
+#title()
+#align(center)[
+  Gidon Rosalki \
+  2026-10-13
+]
+
+#let colourmaths(x, color) = text(fill: color)[$#x$]
+
+= Introduction
+
+Since everyone comes with a different background in mathematics to their masters in CompSci, this course exists to
+ensure everyone knows a standard foundation. \
+The course is split into 4 main parts, Linear Algebra, Infinitesimal Calculus, Optimisation, and Probability. Aside from
+the optimisation section, it will all be content already taught in one's bachelors degree. The purpose is to give us
+skills that we may use in our research.
+
+= Linear Algebra
+
+We will be focusing on slightly less abstract concepts than in the bachelors courses. We will only work over the field
+$RR$, real matrices, and vector spaces, and so on, rather than abstract fields, spotlighting applicative concepts.
+
+== Reminder
+
+=== Matrices
+
+Let there be a matrix $M in "Mat"_(n times m) (RR)$. The vectors will all be column vectors (unless stated otherwise) $v
+in RR^(n)$. We may transpose our matrix $M : M^T in "Mat"_(m times n) (RR)$ defined such that $M^T_(i, j) = M_(j, i)$.
+
+If $M$ is a square matrix: $M in "Mat"_(n times n) (RR)$, it is called *invertible* if $exists M^(-1) in M_(n) (RR) : M
+dot M^(-1) = I_n$.
+
+=== Eigenvalues and Eigenvectors
+
+Let there be $M in "Mat"_(n) (RR)$.
+
+*Definition* (eigenvalue): $lambda in RR$ is called an *eigenvalue* of $M$ if $exists v in RR^n != 0 : M dot v = lambda
+dot v$. The eigenvalues of a matrix are called its _spectrum_.
+
+*Definition* (eigenvector): Given an eigenvalue $lambda$ of matrix $M$, where $M dot v = lambda dot v$, then $v in RR^n
+!= 0$ is called an *eigenvector*.
+
+*Definition* (eigenspace): The *eigenspace* of $lambda$ is defined as: $V_lambda = {v in RR^n : M v = lambda v}$. \
+It is easy to show that $V_lambda$ is a linear subspace of $RR^n$. \
+$dim (V_lambda)$ is the (geometric) multiplicity of $lambda$.
+
+*Fact*:
+$
+  sum_(lambda "eigenvalue of" M)^() dim (V_lambda) <= n
+$
+
+*Definition* (diagonalisable): $M$ is diagonalisable if there is an invertible matrix $P$, and a diagonal matrix $D$
+such that $M = P D P^(-1)$, where $P, D in M_(n)$.
+
+Let $v_i$ be the $i$-th column of $P$. Then
+$
+              M dot v_i & = P D P^(-1) dot v_i \
+  P dot P^(-1) ==> quad & = P D e_i \
+                        & = P dot D_(i i) dot e_i \
+                        & = D_(i, i) P e_i \
+                        & = D_(i, i) v_i \
+$
+This implies that $D_(i, i)$ is an eigenvalue, with eigenvector $v_i$. From this we may conclude:
+
+*Theorem*: $M$ is diagonalisable $<==>$ there is a basis $v_1, dots, v_n$ of $RR^n$ of eigenvectors of $M$.
+
+=== Orthogonal Diagonalisation
+
+We will firstly discuss the standard inner product. There are many different inner products, but we will focus on the
+standard one:
+$
+  x, y in RR^n ==> lr(chevron.l x, y chevron.r) = sum_(i = 1)^(n) x_i dot y_i
+$
+
+Let $0 != v_1, dots, v_n in RR^n$. 
+
+*Definition* (orthogonal): $v_1, dots, v_n$ are _orthogonal_ if $forall i != j quad lr(chevron.l v_i, v_j chevron.r) = 0$
+
+*Definition* (orthonormal): $v_1, dots, v_n$ are _orthonormal_ if they are orthogonal, and \ 
+$forall i quad norm(v_i) = sqrt(lr(chevron.l v_i, v_i chevron.r)) = 1$ (unit vectors)
+
+*Definition* (orthogonal matrix): An $n times n$ matrix $V$ is called _orthogonal_ if $V^T dot V = I$. Equivalent to:
++ The columns of $V$ are orthonormal
++ The rows of $V$ are orthonormal
++ $V^(-1) = V^T$
++ $forall x in RR^n quad norm(V x) = norm(x)$
+
+*Definition* (orthogonally diagonalisable): A matrix $M$ is _orthogonally diagonalisable_ if there exist orthogonal matrix
+$V$, diagonal matrix $D$ such that $M = V dot D dot V^T$.
+
+*Theorem* (Spectral theorem for symmetric matrices): If $M$ is symmetric ($M = M^T$), then $M$ is orthogonally
+diagonalisable, i.e. there is an orthonormal basis of eigenvectors. In other words $M = V dot D dot V ^ (T)$, where $V$
+is orthogonal, and $D$ is diagonal.