1.4. Geometry of the real numbers

Geometry of $\R^n$

A subspace of $\R^n$ is a subset of $\R^n$ such that it is closed under addition and multiplication by scalars. That is, $V \subset \R^n$ is a vector subspace if $\vec x, \vec y \in V$ and $a\in\R$, then $\vec x + \vec y \in V$ and $a\vec x \in V$.

In $\R^2$, this includes

Lines through the origin.
The set of the zero vector ${\vec 0}$.
$\R^2$ itself.

The dot product of two vectors $\vec x \cdot \vec y$ is $\sum_{i=1}^n x_i y_i$.

\[\frac{\vec x \cdot \vec y}{|\vec x||\vec y|} = \cos \theta.\]

The cross product in $\R^3$ of two vectors $\vec x \times \vec y$ is

\[\begin{bmatrix} \det \begin{bmatrix}a_2 & b_2\\a_3&b_3\end{bmatrix}\\ -\det\begin{bmatrix}a_1&b_1\\a_3&b_3\end{bmatrix}\\ \det\begin{bmatrix}a_1&b_1\\a_2&b_2\end{bmatrix} \end{bmatrix}\]

The Cauchy-Schwarz Inequality proves that for two vectors, $|\vec x \cdot \vec y| \leq |\vec x||\vec y|$.

It follows that for matrices, $|AB| \leq |A||B|$.

The triangle inequality proves that $|a+b| \leq |a| - |b|$ since,

\[|a| = |a+b-b|\\ \leq |a+b|+|-b|\\ =|a+b|+|b|.\]

Generalized, this is,

\[\left |\sum_{i=0}^{\infty}x_i\right |\leq \sum_{i=0}^{\infty}|x_i|\]

if $\sum_{i=0}^{\infty} x_i$ converges. This works because the absolute value function is continuous.