Transformations using Matrices

• Linear Transformations: A transformation is a mapping that moves a point $(x, y)$, known as the object, to a new position $(x', y')$, known as the image. In matrix algebra, this is achieved by representing the point as a column vector and multiplying it by a transformation matrix $M$.

• Matrix Representation: To find the image of a point, the matrix $M$ is placed to the left of the column vector. The resulting calculation follows the standard row-by-column multiplication rule: $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix}$

• Vertex Matrices: When transforming a geometric shape, the coordinates of all vertices can be combined into a single vertex matrix where each column represents a point. Multiplying the transformation matrix by this vertex matrix produces a new matrix where each column is the corresponding image vertex.

• The Unit Square Method: Any transformation matrix can be derived by observing the movement of the unit vectors $i = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $j = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. If $i$ moves to $i'$ and $j$ moves to $j'$, the transformation matrix $M$ is simply the concatenation of these two image vectors as columns: $M = (i' | j')$.

• Preservation of the Origin: Because these are linear transformations, the origin $(0,0)$ is always an invariant point. This means that no matter the matrix, the origin will always map to itself, as $M \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$.

• Linearity and Straight Lines: A key property of these transformations is that they map straight lines to straight lines. While the lengths and angles between lines may change, the fundamental linear structure of the geometry is preserved.

Reflections

• Reflection in Axes: Reflecting across the $x$-axis uses the matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, while the $y$-axis uses $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$. These matrices flip the sign of the coordinate perpendicular to the axis of reflection.
• Reflection in $y = \pm x$: Reflecting in $y=x$ swaps the $x$ and $y$ coordinates using $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. Reflecting in $y=-x$ swaps and negates them using $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$.

Rotations

• General Rotation: A rotation anticlockwise by an angle $\theta$ about the origin is defined by the matrix: $M = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$
• Clockwise Rotations: To perform a clockwise rotation, use a negative value for $\theta$. Note that $\cos(-\theta) = \cos \theta$ and $\sin(-\theta) = -\sin \theta$.

Stretches and Enlargements

• Enlargement: An enlargement by scale factor $k$ about the origin is represented by $kI = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$. This scales both dimensions equally.
• Stretch: A stretch parallel to an axis scales only one dimension. A stretch parallel to the $x$-axis by factor $a$ is $\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix}$, while a stretch parallel to the $y$-axis by factor $b$ is $\begin{pmatrix} 1 & 0 \\ 0 & b \end{pmatrix}$.

• Successive Transformations: When applying multiple transformations, the order of matrix multiplication is critical. If transformation $P$ is applied first and $Q$ is applied second, the combined matrix $M$ is calculated as $M = QP$. The first transformation always appears on the right.

• Inverse Transformations: The inverse matrix $M^{-1}$ represents the reverse transformation that maps the image back to the original object. Applying a transformation followed by its inverse results in the identity matrix $I$, which represents no change.

• Inverse of Combined Matrices: To reverse a sequence of transformations, one must reverse each individual step in the opposite order. Algebraically, $(QP)^{-1} = P^{-1}Q^{-1}$.

• Rotation vs. Reflection: Rotations preserve the 'sense' (orientation) of the shape and usually have a determinant of $+1$. Reflections reverse the 'sense' and always have a determinant of $-1$.

• $y=x$ vs. $y=-x$: Reflection in $y=x$ uses the matrix with $1$s on the off-diagonal. Reflection in $y=-x$ uses $-1$s on the off-diagonal. Confusing these is a common source of error.

• The Unit Square Check: If you forget a standard matrix, mentally (or physically) draw the unit square and see where $(1,0)$ and $(0,1)$ land. This is the most reliable way to reconstruct any reflection or rotation matrix.

• Order of Operations: Always write combined transformations from right to left. For 'Transformation A followed by Transformation B', the calculation is $B \times A$. Reversing this order is the most common reason for lost marks.

• Area Problems: When asked to find a missing constant in a matrix given an area change, always use the modulus: $|det(M)| = \text{Scale Factor}$. This often leads to two possible algebraic solutions (one positive, one negative).

• Verification: Check your result by transforming a simple point like $(1,1)$. If the resulting image doesn't match the geometric description (e.g., it's in the wrong quadrant for a rotation), re-check your matrix multiplication.