Lecture Notes

A vector space is a set with certain properties. What is a set?

A set is a collection of elements. For example, we can say, “the set of students in this room”, where each student would be an element of this set. Or we can say, “the set of possible outcomes of tossing a coin”, then it would contain two elements $\{ Head, Tail\}$.

The real number is a set $\R$. It is an infinite set, meaning it contains infinitely many elements.

The rational number is a set $\Q$. It is also infinite. But it is countable, meaning, we can have a bijection between the natural number $\N$, or we can a way of list all the elements in $\Q$ (for example, we can enumerate them according to the size of the denominators $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\frac{2}{3}, \frac{1}{4},\frac{3}{4}, \cdots$

If we are given two sets, denoted as $A$ and $B$, we can define their Cartesian product, this is another set, defined as $A \times B =\{ (a,b) \mid a \in A, b \in B \}$ It consist of all possible pairs in $A$ and $B$. If $A$ and $B$ are finite sets, then $A \times B$ is a finite, and $|A \times B| = |A| \times |B|$, where $|A|$ means the number of elements in $A$, or 'size' of $A$.

We can form consecutive products, like $(A \times B) \times C$. It is obvious that $(A \times B) \times C = A \times (B \times C)$, hence we will sometimes omit the parenthesis, and simply write $A \times B\times C$.

We write $\R^2 = \R \times \R$ and similarly $\R^n$ as Cartesian product of $n$ copies of $\R$.

If $A, B$ are two sets, a map $f$ from $A$ to $B$ is an assignment, that for each $a \in A$, we assign an element $f(a) \in A$. This is denoted as $f: A \to B$.

A real valued function on $A$, is a map $f: A \to \R$.