Below are some cute linear algebra results and proofs cherrypicked from various sources. All the standard hypotheses (on the base field, the size of the matrices, etc.) that make the claims valid are assumed. The list will likely be updated.

Proof. It suffices to show that the kernel of the linear transformation is trivial. If is in the kernel, then for each . Since , this forces to be identically zero. Thus .

Proof. A matrix is diagonalisable iff every Jordan block has size . Since the multiplicity of an eigenvalue in the minimal polynomial corresponds to the size of the largest Jordan block, the result follows.

Proof. Since the nullity of is the geometric multiplicity of the eigenvalue , and the geometric multiplicity of an eigenvalue is at most its algebraic multiplicity, we get .

Fact 7. The number of distinct eigenvalues of is at most .

Proof. The rank of a matrix is the number of non-zero eigenvalues and the nullity is the number of zero eigenvalues, both counted with multiplicity.

be a Jordan block. Let be the standard basis vectors, i.e. the -th component of is . Note the action of on the basis vectors: for each (where we take ).

where is the extended complex plane and the ‘point at infinity’ is defined so that

if then and ;

if then .

The following video gives a very illuminating illustration. The sphere in the video is called the Riemann sphere, which in a sense ‘wraps up’ the extended complex plane into a sphere. Each point on the sphere corresponds to a unique point on the plane (i.e. there is a bijection between points on the extended plane and points on the sphere), with the ‘light source’ being the point at infinity. This bijective correspondence is the main reason for including the point at infinity.

According to the video any Möbius transformation can be generated by the four basic ones: translations, dilations, rotations and inversions:

Translation: ,

Dilation: ,

Rotation: ,

Inversion:

Exercise. Show that any Möbius transformation is some composition of these operations.

The Möbius transformations in fact form a group under composition which acts on . Moreover, we have a surjective homomorphism

.

Möbius transformations exhibit very interesting properties, some of which are:

Proposition 1. Given distinct , there is a unique Möbius map such that

Proof. It is not difficult to work out that the unique is given by

.

Proposition 2. The action of on is sharply triply transitive: if are distinct and are distinct, then there eixsts a unique such that for .

Proof. By proposition 1, there is a unique such that

and a unique such that

.

Then is the unique map satisfying the required property.

The cross-ratio of four distinct points is defined to be the unique such that if is the unique map satisfying

then , i.e.

.

One nice thing about cross-ratios is that they are preserved by Möbius transformations.

Proposition 3. If , then .

Proof. Let such that

.

Then . Likewise, if satisfies

then by proposition 1, so .

From we observe that some permutations of leave the value of the cross-ratio invariant, e.g. is one such. What about the others?

Let act on the indices of the cross-ratio . The permutations that fix form the stabiliser subgroup of of this action. Using transitivity and invariance (propositions 2 and 3), the orbit of is just the different assignments of the values to ; i.e. the distinct cross-ratios that we get by permuting the indices are just

, , ,

, , .

Let such that

.

Then writing shows that the values of the above cross-ratios are (not necessarily in this order—too lazy to work out the precise order)

and they form the subgroup of that fixes the set . This group is isomorphic to .

So there are in fact four permutations such that

and they form a subgroup of that is isomorphic to the Klein four-group