Matrices and linear transformations

Matrix transformation

$$T_A:\mathbf{x}\mapsto A\mathbf{x}$$

$$A(\mathbf{x})=\{Ax:x\in \mathbf{X}\}$$

$\mathbf{x}$ is the set of input

$$A({\lambda }_1{\mathbf{x}}_1+{\lambda }_2{\mathbf{x}}_2)={\lambda }_{1}A{\mathbf{x}}_1+{\lambda }_{2}A{\mathbf{x}}_2$$

commutative diagram

Other example of commutative diagram in group theory:

Axiom: Linear transformation

Linearity

NOTE

A function $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is called a linear transformation if the following linearity axiom holds for all ${\mathbf{x}}_1,{\mathbf{x}}_2\in {\mathbb{R}}^n$ and all ${\lambda }_1,{\lambda }_2\in \mathbb{R}$

$$T({\lambda }_1{\mathbf{x}}_1+{\lambda }_2{\mathbf{x}}_2)={\lambda }_{1}T({\mathbf{x}}_1)+{\lambda }_{2}T({\mathbf{x}}_2)$$

OR

• $T(\mathbf{x}+{\mathbf{x}}^{\prime })=T(\mathbf{x})+T({\mathbf{x}}^{\prime })$
• $T(\lambda \mathbf{x})=\lambda T(\mathbf{x})$

linear functional

a special case of linear transformation where $m=1$ So $T:{\mathbb{R}}^n\to \mathbb{R}$

What is axiom?

NOTE

An axiom is a defining property of a class of mathematical objects. Exactly the objects satisfying the property belong to the class.

Warning

• In axiomatic mathematics, “class” = collection of all objects satisfying certain axioms.
• The class of groups is defined by the group axioms.
  • The class of vector spaces is defined by the vector space axioms.
• In category theory, “class” = a size notion (collection of objects or morphisms), not the same as “axiomatically defined class.”
• “a possibly proper class of objects,” i.e. too large to be a set
• Category Theory Intro > What is a class

Matrix multiplication

The composition of matrix transformations is again a matrix transformation

Matrix transformation

+−⟲

Link to original

$T_a:{\mathbb{R}}^n\to {\mathbb{R}}^b$ $T_b:{\mathbb{R}}^a\to {\mathbb{R}}^n$ $T_c:{\mathbb{R}}^a\to {\mathbb{R}}^b$

Lemma 2.40

Lemma

$$(AB{)}^{\top }=B^{\top }{A}^{\top }$$

Example

$x^{\top }{A}^{\top }Ax=(Ax{)}^{\top }Ax=||Ax|{|}^2$

$\underset{Scalar}{\underbrace{\underset{Covector}{\underbrace{{\mathbf{v}}^{\top }}}\underset{Vector}{\underbrace{\underset{Matrix}{\underbrace{(\mathbf{w}{\mathbf{v}}^{\top })}}\underset{Vector}{\underbrace{w^{\top }}}}}}}$

Characters of matrix multiplication

Lemma 2.42