Basis and linear maps

Vector Spaces

Subspaces definition

Linear Independence

Spanning Sets and Linear Combinations

Basis and Dimension

Definition of a Basis

Dimension of a Vector Space ($\dim (V)$)

Rank of a system of vectors

Linear Maps (Transformations)

Definition: $L(\vec{u}+\vec{v})=L(\vec{u})+L(\vec{v})$ and $L(c\vec{u})=cL(\vec{u})$

Matrix Representation of Linear Maps

Kernel (Null space) and Image (Range)

Rank-Nullity Theorem: $\dim (V)=\text{rank}(T)+\text{nullity}(T)$

Change of Basis

Transition Matrices ($P_{B\to B^{\prime }}$ and $P^{-1}$)

Coordinate Vectors relative to a Basis

Similarity Transformations $A^{\prime }=P^{-1}AP$

Geometric Transformations

2D/3D Rotations

Scaling and Homothety

Shearing

Projections and Reflections

Homogeneous Coordinates (Translation matrices)