Transformations

A transformation is a function.

E.g. $T:ℝ^2\to ℝ^2$ defined by $\begin{bmatrix} x \\ y \end{bmatrix} \to \begin{bmatrix} 2x \\ y \end{bmatrix}$

Image of a Set

Let $L:ℝ^n\to ℝ^m$ be a transformation and let $X\sub ℝ^n$ be a set.

The image of the set $X$ under $L$:

$L(X)=\{\vec{y}\in ℝ^m:\vec{y}=L(\vec{x}) \text{ for some } \vec{x} \in X\}$

Transformation Properties

• $L: ℝ^n\to ℝ^m$ has domain $R^n$ and codomain $R^m$
• $L: ℝ^n\to ℝ^m$ is one-to-one iff $\forall\vec{y} \in \mathbb{R}^m,T(\vec{x})=\vec{y}$ has at most one solution $\vec{x} \in \mathbb{R}^n$
• $L: ℝ^n\to ℝ^m$ is onto iff $\forall \vec{y}\in \mathbb{R}^m,\exists \vec{x} \in ℝ^n$ s.t. $T(\vec{x})=\vec{y}$
• $L$ is not a matrix nor a set.
• $L$ cannot contain any vectors, but $L(X)$ can.

Linear Transformations

Let $V$ and $W$ be subspaces. $T:V\to W$ is a linear transformation if

$T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$ and $T(a\vec{v}) = aT(\vec{v})$

$\forall \vec{u},\vec{v}\in V,\forall a \in \mathbb{R}$

Linear Transformation Properties

• $T$ can be one-to-one (TODO: Are they????)
  • $T(\vec{x})=0$ has only the trivial solution $\vec{x}=0$
  • If $M$ is the standard matrix of $T$, the columns of $M$ are linearly independent.
• $T$ is onto (TODO: Are they???)

  • $T(\vec{x})=\vec{y}$ has solutions for all $\vec{y} \in \mathbb{R}^m$

  • If $M$ is the standard matrix of $T$, the columns of $M$ spans $ℝ^m$

    (Every $\vec{v}\in \mathbb{R}^m$ is a linear combination of the columns of $M$)