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Represent and apply transformations with matrices

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Which representation should be built first for a sequence of plane transformations?

Year 12, VCE Units 3 & 4, Chapter 3, 4 difficulty levels, No calculatorBuild a worksheet

Using matrices for plane transformations

A plane transformation can be written as a linear matrix followed by a translation vector. For a sequence, write the first-acting matrix on the right. For an invertible coordinate map, solve for the original coordinates when finding an image curve

\[\begin{bmatrix}x'\\y'\end{bmatrix}=M\begin{bmatrix}x\\y\end{bmatrix}+\begin{bmatrix}t_x\\t_y\end{bmatrix}\]
  1. Write the affine map Represent a linear transformation by a matrix and add the translation vector separately \(\displaystyle \begin{bmatrix}x'\\y'\end{bmatrix}=M\begin{bmatrix}x\\y\end{bmatrix}+\begin{bmatrix}t_x\\t_y\end{bmatrix}\)
  2. Match each axis to its row The first row controls x' and the second row controls y', so diagonal entries must be placed on the matching axes \(\displaystyle M=\begin{bmatrix}2&0\\0&-1\end{bmatrix}\quad\Rightarrow\quad x'=2x,\ y'=-y\)
  3. Apply transformations in order For a sequence of linear transformations, write the first-acting matrix on the right. Each subsequent matrix multiplies the product on the left \(\displaystyle M_1\text{ then }M_2\quad\Rightarrow\quad M=M_2M_1\)
  4. Solve for a curve image For an invertible coordinate map, solve for the original variables before substituting into the curve equation \(\displaystyle x'=2x+1\quad\Rightarrow\quad x=\frac{x'-1}{2}\)

Worked example

A point (3,2) is first dilated in the x-direction by factor 2 and then translated by (1,−3). Find the affine map and the image point

\[M_1=\begin{bmatrix}2&0\\0&1\end{bmatrix},\quad t_1=\begin{bmatrix}0\\0\end{bmatrix},\quad M_2=\begin{bmatrix}1&0\\0&1\end{bmatrix},\quad t_2=\begin{bmatrix}1\\-3\end{bmatrix}\]
  1. \(\displaystyle M_1=\begin{bmatrix}2&0\\0&1\end{bmatrix},\quad t_1=\begin{bmatrix}0\\0\end{bmatrix}\)The first transformation doubles x and leaves y unchanged
  2. \(\displaystyle M_2=\begin{bmatrix}1&0\\0&1\end{bmatrix},\quad t_2=\begin{bmatrix}1\\-3\end{bmatrix}\)The second transformation is the translation by (1,−3)
  3. \(\displaystyle M=M_2M_1=\begin{bmatrix}2&0\\0&1\end{bmatrix},\quad t=t_2=\begin{bmatrix}1\\-3\end{bmatrix},\quad x'=2x+1,\quad y'=y-3\)The identity matrix leaves the first matrix unchanged, and the translation is added after the dilation
  4. \(\displaystyle \begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}2&0\\0&1\end{bmatrix}\begin{bmatrix}3\\2\end{bmatrix}+\begin{bmatrix}1\\-3\end{bmatrix}=\begin{bmatrix}7\\-1\end{bmatrix}\)Applying the affine map sends (3,2) to (7,−1)

Common mistakes

Putting a scale on the wrong axis
✕\(\displaystyle M=\begin{bmatrix}1&0\\0&2\end{bmatrix}\quad\text{for an x-dilation by factor }2\)✓\(\displaystyle M=\begin{bmatrix}2&0\\0&1\end{bmatrix}\quad\text{for an x-dilation by factor }2\)
The first matrix row determines the image x-coordinate and the second row determines the image y-coordinate
Translating before scaling
✕\(\displaystyle x'=2(3+1)=8\)✓\(\displaystyle x'=2(3)+1=7\)
The linear transformation acts first and the translation is added afterwards, so the translation is not multiplied by the scale
Reversing the preimage calculation
✕\(\displaystyle x'=2x+1\quad\Rightarrow\quad x=2x'+1\)✓\(\displaystyle x'=2x+1\quad\Rightarrow\quad x=\frac{x'-1}{2}\)
An image curve needs the original coordinate written in terms of the image coordinate, so x must be expressed in terms of x'

Difficulty levels

introApply a single matrix or affine transformation to a point
standardCompose a sequence of transformations and find an image point or curve
challengeReverse a coordinate map or determine parameters from a transformed object
examCombine matrix products, translations and curve equations in an exact transformation problem

Questions about this topic

Why does matrix order matter?+

Matrix multiplication represents composition, so the rightmost matrix acts first on a column vector

Which matrix entry controls an x-dilation?+

For a diagonal coordinate matrix, the top-left entry multiplies x and the bottom-right entry multiplies y.

Why is a curve image found using the inverse coordinate map?+

For an invertible coordinate map, express the original coordinates using x' and y' before substituting into the original curve equation

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 3: Functions, transformations and inverses

In this chapter

Form and analyse composite functionsInvestigate functional equationsApply and describe transformations of a functionRepresent and apply transformations with matricesForm and analyse an inverse relation or functionComposition with an inverse and intersections of a function and its inverse