On Algebra

Table of Contents

Essence of Linear Algebra

Let us talk about the essence of Linear Algebra.

Span and Basis

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

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

Cross Product

Change of Basis

Eigenvectors and Eigenvalues

Av=λv    (AλI)v=0    given that v0,we have the necessity and sufficiency of det(AλI)=0A\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

About Linear Transformations