Using Matrices with Transformations
Use matrices to describe transformations for Victorian Year 11 Mathematical Methods (VCAA) — two-by-two matrices that dilate, reflect or otherwise move the points of the plane.
You will learn to write the matrix for a dilation or reflection, find the image of a point or shape by matrix multiplication, and identify the transformation that a given matrix produces.
Theory
In Year 11 Mathematical Methods (Unit 2), a dilation or reflection of the plane can be written as a \(2\times 2\) transformation matrix. Multiplying that matrix by the column vector \(\begin{bmatrix}x\\y\end{bmatrix}\) of a point gives the image point. This page shows how to apply a transformation matrix, recognise the reflection and dilation matrices, and work backwards to find an unknown factor.
A point \((x,\,y)\) is written as the column vector \(\begin{bmatrix}x\\y\end{bmatrix}\). A linear transformation such as a dilation or reflection is a \(2\times 2\) matrix \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\); the image of the point is the product \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}\).
A reflection flips a point across a line (the \(x\)-axis, the \(y\)-axis, or the line \(y=x\)). A dilation stretches distances from an axis by a constant factor: a dilation of factor \(k\) from the \(x\)-axis multiplies every \(y\)-coordinate by \(k\), and a dilation of factor \(k\) from the \(y\)-axis multiplies every \(x\)-coordinate by \(k\).
The image of \((x,\,y)\) under a \(2\times 2\) matrix is the matrix product:
The standard reflection and dilation matrices for Units 1 & 2 are:
| Transformation | Matrix | Effect on \((x,y)\) |
|---|---|---|
| Reflection in the \(x\)-axis | \(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\) | \((x,\,-y)\) |
| Reflection in the \(y\)-axis | \(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\) | \((-x,\,y)\) |
| Reflection in \(y=x\) | \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\) | \((y,\,x)\) |
| Dilation factor \(k\) from the \(x\)-axis | \(\begin{bmatrix}1&0\\0&k\end{bmatrix}\) | \((x,\,ky)\) |
| Dilation factor \(k\) from the \(y\)-axis | \(\begin{bmatrix}k&0\\0&1\end{bmatrix}\) | \((kx,\,y)\) |
| Dilation factor \(k\) from both axes | \(\begin{bmatrix}k&0\\0&k\end{bmatrix}\) | \((kx,\,ky)\) |
How to apply a transformation matrix
- Write the point as a column vector \(\begin{bmatrix}x\\y\end{bmatrix}\).
- Multiply the matrix by the vector: top image entry is \(ax+by\), bottom is \(cx+dy\). Show each product.
- Read off the image \((x',\,y')\) from the two entries of the result.
To identify or reverse a transformation
- Identify: multiply the matrix by a general point \(\begin{bmatrix}x\\y\end{bmatrix}\) and describe the effect (e.g. \((x,-y)\) is a reflection in the \(x\)-axis).
- Reverse: set the image entries equal to the given image coordinates and solve the resulting equations for the unknown factors.
Write the point as a column vector and multiply:
| \(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\begin{bmatrix}4\\-2\end{bmatrix}\) | \(=\) | \(\begin{bmatrix}(-1)(4)+(0)(-2)\\(0)(4)+(1)(-2)\end{bmatrix}\) |
| \(=\) | \(\begin{bmatrix}-4\\-2\end{bmatrix}\) |
The \(x\)-coordinate changes sign; the \(y\)-coordinate is unchanged.
The image point is \((-4,\,-2)\).
Multiply the matrix by the column vector:
| \(\begin{bmatrix}3&0\\0&2\end{bmatrix}\begin{bmatrix}5\\4\end{bmatrix}\) | \(=\) | \(\begin{bmatrix}(3)(5)+(0)(4)\\(0)(5)+(2)(4)\end{bmatrix}\) |
| \(=\) | \(\begin{bmatrix}15\\8\end{bmatrix}\) |
Read off each image coordinate:
| \(x'\) | \(=\) | \(15\) |
| \(y'\) | \(=\) | \(8\) |
The image of the corner is \((15,\,8)\).
Apply the reflection matrix:
| \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}3\\4\end{bmatrix}\) | \(=\) | \(\begin{bmatrix}(0)(3)+(1)(4)\\(1)(3)+(0)(4)\end{bmatrix}\) |
| \(=\) | \(\begin{bmatrix}4\\3\end{bmatrix}\) |
Reflecting in \(y=x\) swaps the two coordinates.
The image point is \((4,\,3)\).
Set up the image vector:
| \(\begin{bmatrix}a&0\\0&b\end{bmatrix}\begin{bmatrix}3\\5\end{bmatrix}\) | \(=\) | \(\begin{bmatrix}3a\\5b\end{bmatrix}=\begin{bmatrix}12\\15\end{bmatrix}\) |
Equate the \(x\)-entries and solve for \(a\):
| \(3a\) | \(=\) | \(12\) |
| \(a\) | \(=\) | \(4\) |
Equate the \(y\)-entries and solve for \(b\):
| \(5b\) | \(=\) | \(15\) |
| \(b\) | \(=\) | \(3\) |
The dilation factors are \(a=4\) and \(b=3\).
Common pitfalls
Frequently asked questions
How do you use a matrix to transform a point?
Write the point as a column vector \(\begin{bmatrix}x\\y\end{bmatrix}\), multiply the \(2\times 2\) matrix on its left, and read the image \((x',y')\) from the two entries of the result.
Which matrix is a reflection in the x-axis?
\(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\). It keeps \(x\) and changes the sign of \(y\), sending \((x,y)\) to \((x,-y)\).
Which matrix is a reflection in the y-axis?
\(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\), which sends \((x,y)\) to \((-x,y)\). Reflection in \(y=x\) is \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\), which swaps them to \((y,x)\).
How do you write a dilation as a matrix?
A dilation of factor \(k\) from the \(x\)-axis is \(\begin{bmatrix}1&0\\0&k\end{bmatrix}\); from the \(y\)-axis it is \(\begin{bmatrix}k&0\\0&1\end{bmatrix}\); from both axes it is \(\begin{bmatrix}k&0\\0&k\end{bmatrix}\).
How do you identify the transformation a matrix represents?
Multiply the matrix by a general point \(\begin{bmatrix}x\\y\end{bmatrix}\) and describe the image, for example \((x,-y)\) is a reflection in the \(x\)-axis and \((2x,2y)\) is a dilation of factor \(2\) from both axes.
How do you find an unknown dilation factor?
Multiply the matrix by the original point, set each image entry equal to the given image coordinate, and solve for the unknown factor.