LINEAR TRANSFORMATIONS

Transformations

• A transformation T from Rⁿ to R^m takes ${\bf x}\in R^n$ to ${\bf p}\in R^m$.
• Notation: $T({\bf x}) = {\bf p}$; $T: R^n \rightarrow R^m$.
• Rⁿ is the domain of T and R^m is the codomain.
• If $T({\bf x}) = {\bf p}$, ${\bf p}$ is the image of ${\bf x}$ using T.
• The set of all images is the range of T.

Matrix Transformations

If A is an $m \times n$ matrix, the transformation $T({\bf x}) = A{\bf x}$has domain Rⁿ and codomain R^m.

Linear Transformations

• A transformation T is linear if:

  1.

  $T({\bf u}+{\bf v}) = T({\bf u}) + T({\bf v})$ for ${\bf u}, {\bf v}$ in the domain of T.

  2.

  $T(c{\bf u}) = cT({\bf u})$ for all ${\bf u}$ in the domain of T and scalars c.

• Matrices are linear transformations.
• Additional properties:
  $T({\bf0}) = {\bf0}$;
  $T(c{\bf u}+ d{\bf v}) = cT({\bf u}) + dT({\bf v})$;
  $T(c_1{\bf u}_1 + c_2{\bf u}_2 + \cdots + c_p{\bf u}_p ) = c_1T({\bf u}_1) + c_2T({\bf u}_2) + \cdots + c_pT({\bf u}_p)$.

Geometry applications:

Shears, rotations, dilations, contractions

The Matrix of a Linear Transformation:

linear transformations can be represented by matrices:
if $T: R^n \rightarrow R^m$ and T is linear, then $T({\bf x}) = A{\bf x}$ with

$$A = [ T({\bf e}_1), T({\bf e}_2), \ldots, T({\bf e}_n) ].$$

Onto and One-To-One

• Defintion: a transformation $T: R^n \rightarrow R^m$ is onto R^m if, for every ${\bf b}\in R^m$, there exists at least one ${\bf x}\in R^n$ with $T({\bf x}) = {\bf b}$.
• Defintion: a transformation $T: R^n \rightarrow R^m$ is one-to-one if, for each ${\bf b}\in R^m$, there is at most one ${\bf x}\in R^n$ with $T({\bf x}) = {\bf b}$.
• Theorem: a transformation $T: R^n \rightarrow R^m$ is one-to-one iff the equation $T({\bf x}) = {\bf0}$ has only the trivial solution.
• Theorem: if $T: R^n \rightarrow R^m$ with standard matrix A:
  • T is onto iff the columns of A span R^m.
  • T is one-to-one iff the columns of A are linearly independent.