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. We have . By induction, for any . Hence the result follows.
Fact 2. Let be distinct. Then the Vandermonde matrix
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.
Fact 4. If and , then are linearly independent.
Proof. Note that . Let and suppose that . Multiplying both sides by for shows that for all , as desired.
Corollary 1. If , then for .
Proof. If , then .
Corollary 2. If is , and , then for each . In particular, is similar to the nilpotent Jordan block of size .
Fact 5. If is linear on , then .
Proof. The short exact sequence
Corollary. If is a subspace, then .
Fact 6. If is , then , where is the algebraic multiplicity of the eigenvalue of .
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.