Algebra

Algebra

1. Linear systems and matrices

   • (R)REF
     • Definition 1: matrix A is in REF if (i) all zero rows are at the bottom, (ii) first non-zero entry in each non-zero row is 1, (iii) the pivot in row i + 1 is strictly to the right of the pivot in row i

2. Vector spaces and linear maps

   • Linear subspace
     • Written U≤V
     • 0 ∈ U
     • ∀ λ, μ ∈ F, v, w ∈ U, λv + μw ∈ U
   • Let ψ:V->W
     • Kernel
       • Ker ψ = {v in V : ψ(v) = 0}
       • Subspace of V
     • Image
       • Im ψ = {w in W : w = ψ(v) for some v in V}
       • Im ψ = {ψ(v) : v in V}
       • Subspace of W
   • Linear span
     • For a subspace S of V, <S> is the smallest subspace of V containing S
     • A list of vectors v1, ... , vn of vectors in V is

3. Dimension and rank

4. Determinants and inverses

5. Eigenvalues and diagonalism