Matrix Transformations (DP IB Applications & Interpretation (AI): HL): Revision Note

Transformation by a matrix

What is a transformation matrix?

• A transformation maps an object to its image

  • e.g. translations, rotations, reflections, stretches, enlargements, etc

• Matrices can be used to describe the transformation

  • They are called transformation matrices

• In this course, the matrix transformations will be of the form

$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}e\\ f\end{array}\right)$

• The matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is normally labelled $\mathit{T}$

How do I find the coordinates of an image after a transformation?

• STEP 1
  Write the coordinates of the vertices of the original object as column vectors

  • e.g. (0, 1), (0, 2) and (3, 1) are written as $\left(\begin{array}{c}0\\ 1\end{array}\right)$, $\left(\begin{array}{c}0\\ 2\end{array}\right)$ and $\left(\begin{array}{c}3\\ 1\end{array}\right)$

• STEP 2
  Substitute each column vector into the matrix transformation

  • e.g. using $\left(\begin{array}{cc}1& 0\\ -3& 1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}1\\ 1\end{array}\right)$

    • $\left(\begin{array}{cc}1& 0\\ -3& 1\end{array}\right)\left(\begin{array}{c}0\\ 1\end{array}\right)+\left(\begin{array}{c}1\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ 2\end{array}\right)$

    • $\left(\begin{array}{cc}1& 0\\ -3& 1\end{array}\right)\left(\begin{array}{c}0\\ 2\end{array}\right)+\left(\begin{array}{c}1\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ 3\end{array}\right)$

    • $\left(\begin{array}{cc}1& 0\\ -3& 1\end{array}\right)\left(\begin{array}{c}3\\ 1\end{array}\right)+\left(\begin{array}{c}1\\ 1\end{array}\right)=\left(\begin{array}{c}4\\ -7\end{array}\right)$

• STEP 3
  Write the column vectors as coordinates

  • e.g. (0,1) ⇾ (1, 2), (0,2) ⇾ (1, 3) and (3,1) ⇾ (4, -7)

How do I find the coordinates of the original point given the image under a transformation?

• You might be asked to find the original coordinates given

  • a matrix transformation $\mathit{T}\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}e\\ f\end{array}\right)$

  • and the coordinates of an image $\left(x\text{'},y\text{'}\right)$

• You just rearrange $\mathit{T}\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}e\\ f\end{array}\right)=\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)$ to find $\left(x,y\right)$

  • Subtract the column vector

    • $\mathit{T}\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}x\text{'}-e\\ y\text{'}-f\end{array}\right)$

  • Pre-multiply by the inverse of $\mathit{T}$

    • $\left(\begin{array}{c}x\\ y\end{array}\right)={\mathit{T}}^{-1}\left(\begin{array}{c}x\text{'}-e\\ y\text{'}-f\end{array}\right)$

• For example, given $\left(\begin{array}{cc}1& 0\\ -3& 1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}1\\ 1\end{array}\right)=\left(\begin{array}{c}3\\ 0\end{array}\right)$

  • $\left(\begin{array}{cc}1& 0\\ -3& 1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}2\\ -1\end{array}\right)$

  • $\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 3& 1\end{array}\right)\left(\begin{array}{c}2\\ -1\end{array}\right)=\left(\begin{array}{c}2\\ 5\end{array}\right)$

  • (2, 5) ⇾ (3, 0)