Linear Transformations

A function $L:R^n\to R^m$ is called a linear transformation if:

$$L(s\vec{x}+t\vec{y})=sL(\vec{x})+tL(\vec{y})$$

• The function preservers linear combinations.

• Property $L(s\vec{x})=sL(\vec{x})$ is called homogeneity.

• Property $L(\vec{x}+\vec{y})=L(\vec{x})+L(\vec{y})$ is called superposition.

• Since every matrix transformation preserves linear combinations, every matrix transformation is a linear transformation.

• A linear tf is a matrix tf it it can be represented as a matrix-vector product and Theorem 32.2 - every linear transformation has a corresponding matrix transformation

• I.e. a tf is linear iff it is also a matrix tf.