Linear Transformations in two dimensions
Linear Transformations: True or False?
- We can represent a translation using a matrix. FALSE. (Matrices can only be used for multiplying points)
- We can convert a 3d coordinate into a 2d coordinate. TRUE. If we set the Z componant to 0 then it will become a 2d point.
Reversing linear transformations
- \(\begin{pmatrix}? & ? \\ ? & ?\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}2y+x \\ 3x\end{pmatrix}\)
- \(\begin{pmatrix}1x+2y \\ 3x+0y\end{pmatrix}\)
- \(\begin{pmatrix}1 & 2 \\ 3 & 0\end{pmatrix}\)
Carrying out linear transformations
\(\begin{pmatrix}-1 & 2 \\ 2 & 1\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}=\mathbf{S}\)
- \((1, 1) \to (1, 3 )\)
- \((3, 1)\to(-1,7)\)
- \((3,3)\to(3, 9)\)
- \((1, 3)\to(5, 5)\)
Determining a Matrix for a specific transformations
- Reflection in the y axis: \(\begin{pmatrix}-1 & 0 \\ 0 & 1\end{pmatrix}\)
- Reflection in the x axis: \(\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\)
- Rotation of 90\(^{\circ}\) about the origin clockwise: \(\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}\)
- Rotation of 90\(^{\circ}\) about the origin anticlockwise: \(\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\)
- Rotation of 180\(^{\circ}\):\(\begin{pmatrix}-1 & 0 \\0 & -1\end{pmatrix}\)
- Reflection in the line \(y=x\): \(\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\)
- Reflection in the line \(y=-x\): \(\begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}\)
- Enlargement of 2:\(\begin{pmatrix}2 & 0 \\ 0 & 2\end{pmatrix}\)
Unit Vectors
- The two unit vectors used are (0, 1) and (1, 0)
- These are labelled as \(i\) and \(j\)
- You can take a shortcut by using unit vectors.
- As shown by this image, if we rotate the unit vector (0, 1) 90\(^\circ\), then we get the new point, (1, 0)
- Combining these two vectors gives us the matrix we need to transform the point.
- \(\begin{pmatrix}0 \\ 1\end{pmatrix}+\begin{pmatrix}1 \\ 0\end{pmatrix}=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\)