Matrix Transformation

The matrix transformation for a matrix $A$ is defined by the function $f_A(\vec{x})=A\vec{x}$ Note:

• For $A\in M_{m\times n}(\mathbb{R})$, we have that $f_A:{\mathbb{R}}^n\to {\mathbb{R}}^m$
  • Notice how the matrix is $m\times n$, but the function is a mapping from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$, because think how Matrix multiplication works

Example

Let $A=\left[\begin{array}{ccc}1& 2& 3\\ 1& -1& 1\end{array}\right]$. Then $A\in M_{2\times 3}(\mathbb{R})$ and $f_A:{\mathbb{R}}^3\to {\mathbb{R}}^2$. We can compute the matrix transformation to vector $\left[\begin{array}{c}1\\ 1\\ 4\end{array}\right]$ by the following: $f_A(1,1,4)=\left[\begin{array}{ccc}1& 2& 3\\ 1& -1& 1\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 4\end{array}\right]=(15,4)$

Linear Transformation are usually written in Matrix form, hence doing Matrix Transformation.

Some Transformations

• Euclidean Transformation using a matrix from the Special Euclidean Group
• Similarity Transformation
• Affine Transformation
• Perspective Transformation

Some transformations are cool because they maintain some property of the vectors that are applied on them. Rotation is a transformation. Translation can be written as a matrix transformation (use the Euclidean Transformation with identity matrix for rotation)!

\begin{array}{c|c|c|c} \hline \text{Transform Name} & \text{Matrix Form} & \text{Degrees of Freedom} & \text{Invariance} \\ \hline \text{Euclidean} \rule{0pt}{20 pt} & \left[ { \begin{array}{*{20}{c}} \mathbf{R} & \mathbf{t}\\ {{\mathbf{0}^T}}&1 \end{array}} \right] & 6 & \text{Length, angle, volume} \\ \text{Similarity} \rule{0pt}{20 pt}& \left[ { \begin{array}{*{20}{c}} {s \mathbf{R}}& \mathbf{t}\\ {{ \mathbf{0}^T}}&1 \end{array}} \right] & 7 & \text{volume ratio} \\ \text{Affine} \rule{0pt}{20 pt}& \left[ { \begin{array}{*{20}{c}} \mathbf{A} & \mathbf{t}\\ {{\mathbf{0}^T}} & 1 \end{array}} \right] & 12 & \text{Parallelism, volume ratio} \\ \text{Perspective} \rule{0pt}{20 pt} & \left[ { \begin{array}{*{20}{c}} \mathbf{A} & \mathbf{t}\\ {{\mathbf{a}^T}} & v \end{array} } \right] & 15 & \text{Plane intersection and tangency} \rule{0pt}{20 pt}\\ \hline \end{array}