Linear Transformations

Definition

If the operation preserves vector addition and scalar multiplication, it can be viewed as a linear transformation, and matrices associated with such operations are often used to represent linear transformations.

To be considered a linear transformation, a matrix operation must satisfy two fundamental properties: additivity and homogeneity.

A linear transformation $T:V\to W$ between vector spaces $V$ and $W$ must satisfy the following for all vectors $\mathbf{u}$ and $\mathbf{v}$ in $V$ and scalars $c$:

1. Additivity: $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$

2. Homogeneity: $T(c\mathbf{u})=cT(\mathbf{u})$

Matrix Addition

Matrix-Scalar Addition 

test

Matrix-Matrix Addition

Matrix Multiplication

Matrix Transpose

Matrix-Matrix Multiplication

Scalar-Matrix Multiplication

Hadamard Product