- Span $=$ defined by all vectors that can be represented as a linear combination of the given vectors.
- Basis of a space $= $ linearly independent vectors that span the entire given space.

Dot product $=$ duality of a linear transformation (use for projections).

$$A\vec{v} = \lambda\vec{v}\\ \implies (A - \lambda I)\vec{v} = 0\\ \implies \text{given that}\ \vec{v} \neq 0,\\ \text{we have the necessity and sufficiency of}\ \det{(A - \lambda I)} = 0$$