Essence of Linear Algebra

Span and Basis

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Span ===﻿ defined by all vectors that can be represented as a linear combination of the given vectors.

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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

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

Linear Transformation