Euclidian transformations, lengths and angles

Linear metrics


We defined a norm as:


A metric is the distance between two vectors.


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:


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



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.


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

Note also that:


Is the Euclidian norm.

Taxicab norm

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




To linear forms


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.


If we are using the real field this is:


Pythagoras’ theorem

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


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

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



Recap: Cauchy-Schwarz inequality

This states that:

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


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


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

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

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

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