Subsection 5.2.1 Transformations
Definition 5.2.1.
A transformation (or mapping) \(T\) form \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is a rule which assigns to each vector \(\vec{x} \in \mathbb{R}^n\) a vector \(T(\vec{x})\) in \(\mathbb{R}^m\text{.}\)
The set \(\mathbb{R}^n\) is called the domain of \(T\) (more generally, a subset of \(\mathbb{R}^n\text{,}\) denoted by \(D(T)\)), and \(\mathbb{R}^m\) is called the codomain of \(T\)
For \(\vec{x} \in \mathbb{R}^n\text{,}\) the vector \(T(\vec{x})\) is called the image of \(\vec{x}\text{.}\)
The range of \(T\) is the set of all images \(T(\vec{x})\text{.}\)
The term “transformation” or “mapping” is used, however the concept is basically the same as the familiar function.
Intuitively, a transformation \(T\) can be understood in terms of the “movement” of mapping one vector to another. More generally, \(T\) maps its domain \(\mathbb{R}^n\) to its range, and this can be thought of as moving all possible input vectors to their corresponding output vectors. In particular, in \(\mathbb{R}^2\) can be thought of as a transformation in the sense of geometry, a way of changing the shape of some objects in some way (translation, rotation, stretching, expansion and compression, etc.).
If \(A\) is an \(m \times n\) matrix, then matrix multiplication by \(A\) is a transformation. In particular, the map \(T(\vec{x}) = A\vec{x}\) is a transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\text{.}\)
Subsection 5.2.2 Linear Transformations
In linear algebra, we only consider transformations with a particular “linear” property, called “linear transformations”. Recall that if \(A\) is an \(m \times n\) matrix, and \(\vec{x}, \vec{y} \in \mathbb{R}^n\text{,}\) then,
\begin{equation*}
A(\vec{x} + \vec{y}) = A\vec{x} + A\vec{y} \quad \text{and} \quad A(c\vec{x}) = cA\vec{x}
\end{equation*}
These properties together make matrix multiplication a so-called “linear” transformation.
Definition 5.2.2.
A transformation \(T\) is linear if,
\(T(\vec{x} + \vec{y}) = T(\vec{x}) + T(\vec{y})\) for all \(\vec{x}, \vec{y} \in D(T)\text{.}\)
\(T(c \vec{x}) = c T(\vec{x})\) for all \(c \in \mathbb{R}, \vec{x} \in D(T)\text{.}\)
Intuitively, property 2 means that if you scale the input by a factor of \(c\text{,}\) then the output will be correspondingly scaled by \(c\text{.}\)
Linear transformations are said to be operation preserving, in particular preserving the operations of addition and scalar multiplication. This is because for a linear map, adding \(\vec{x} + \vec{y}\) in \(\mathbb{R}^n\) and then applying \(T\) is equivalent to applying \(T\) to \(\vec{x}\) and \(\vec{y}\) and then adding their result in \(\mathbb{R}^m\text{.}\)
With this definition, every matrix transformation is a linear transformation.
Theorem 5.2.3. Maps 0 to 0.
If \(T\) is a linear transformation, then \(T(\vec{0}) = \vec{0}\text{.}\)
Proof.
By linearity, \(T(\vec{0}) = T(0\vec{x}) = 0 \cdot T(\vec{x}) = \vec{0}\text{.}\)
Geometrically, a transformation is linear if it maps all (straight) lines to lines, and fixes the origin. Linear transformations have the property that grid lines remain parallel and evenly spaced.
Theorem 5.2.4. General linear property.
Let \(T\) be a linear transformation, \(\vec{v}_1, \dots, \vec{v}_n \in D(T), a_1, \dots, a_n \in \mathbb{R}\text{.}\) Then,
\begin{equation*}
\boxed{T(a_1 \vec{v}_1 + \dots + a_n \vec{v}_n) = a_1 T(\vec{v}_1) + \dots + a_n T(\vec{v}_n)}
\end{equation*}
More compactly,
\begin{equation*}
T\brac{\sum_{k=1}^n a_k \vec{v}_k} = \sum_{k=1}^n a_k T(\vec{v}_k)
\end{equation*}
