Euclidian transformations, lengths and angles

Linear metrics

Metrics

We defined a norm as:

\(||v||=v^TMv\)

A metric is the distance between two vectors.

\(d(u,v)=||u-v||=(u-v)^TM(u-v)\)

Metric space

A set with a metric is a metric space.

Inducing a topology

Metric spaces can be used to induce a topology.

Translation symmetry

The distance between two vectors is:

\((v-w)^TM(v-w)\)

So what operations can we do now?

As before, we can do the transformations which preserve \(u^TMv\), such as the orthogonal group.

But we can also do other translations

\((v-w)TM(v-w)\)

\(v^TMv+w^TMw-v^TMw-w^TMv\)

so symmetry is now \(O(3,1)\) and affine translations

Translation matrix

\([[1,x][0, 1]]\) moves vector by \(x\).

Specific groups

The affine group

The Euclidian group

The Galilean group

The Poincaré group

Non-linear norms

\(L_p\) norms (\(p\)-norms)

\(L^P\) norm

This generalises the Euclidian norm.

\(||x||_p=(\sum_{i=1}^{n}|x|^p_i)^{1/p}\)

This can defined for different values of \(p\). Note that the absolute value of each element in the vector is used.

Note also that:

\(||x||_2\)

Is the Euclidian norm.

Taxicab norm

This is the \(L^1\) norm. That is:

\(||x||_1=\sum_{i=1}^{n}|x|_i\)

Angles

Cauchy-Schwarz

To linear forms

Norms

We can use norms to denote the “length” of a single vector.

\(||v||=\sqrt {\langle v, v\rangle }\)

\(||v||=\sqrt {v^*Mv}\)

Euclidian norm

If \(M=I\) we have the Euclidian norm.

\(||v||=\sqrt{v^*v}\)

If we are using the real field this is:

\(||v||=\sqrt{\sum_{i=1}^{n}v^2_i}\)

Pythagoras’ theorem

If \(n=2\) we have in the real field we have:

\(||v||=\sqrt{v_1^2+v_2^2}\)

We call the two inputs \(x\) and \(y\), and the length \(z\).

\(z=\sqrt {x^2+y^2}\)

\(z^2=x^2+y^2\)

Angles

Recap: Cauchy-Schwarz inequality

This states that:

\(|\langle u,v\rangle |^2 \le \langle u, u\rangle \dot \langle v, v\rangle \)

Or:

\(\langle v,u\rangle\langle u,v\rangle \le \langle u, u\rangle \dot \langle v, v\rangle \)

Introduction

\(\langle v,u\rangle\langle u,v\rangle \le \langle u, u\rangle \dot \langle v, v\rangle \)

\(\dfrac{\langle v,u\rangle\langle u,v\rangle}{||u||.||v||} \le ||u||.||v||\)

\(\dfrac{||u||.||v||}{\langle v,u\rangle} \ge \dfrac{\langle u,v\rangle}{||u||.||v||}\)

\(\cos (\theta )=\dfrac{\langle u,v\rangle }{||u||.||v||}\)