Matrix transformations about the origin

Reflection in the x-axis

We can represent a reflection in the x-axis using the matrix:

\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}1&0\\0&-1\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}1&0\\0&-1\end{pmatrix}
• So:
  • A = \begin{pmatrix}1&0\\0&-1\end{pmatrix}
• Answer: \begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}

Reflection in the y-axis

We can represent a reflection in the y-axis using the matrix:

\begin{pmatrix}-1 & 0\\0 & 1\end{pmatrix}

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}-1&0\\0&1\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}-1&0\\0&1\end{pmatrix}
• So:
  • A = \begin{pmatrix}-1&0\\0&1\end{pmatrix}
• Answer: \begin{pmatrix}-1 & 0\\0 & 1\end{pmatrix}

Reflection in the line y = x

We can represent a reflection in the line y = x using the matrix:

\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}0&1\\1&0\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}0&1\\1&0\end{pmatrix}
• So:
  • A = \begin{pmatrix}0&1\\1&0\end{pmatrix}
• Answer: $\begin{pmatrix}0 & 1\1 & 0\end{pmatrix}

Reflection in the line y = -x

We can represent a reflection in the line y = -x using the matrix:

\begin{pmatrix}0 & -1\\-1 & 0\end{pmatrix}

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}0&-1\\-1&0\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}0&-1\\-1&0\end{pmatrix}
• So:
  • A = \begin{pmatrix}0&-1\\-1&0\end{pmatrix}
• Answer: \begin{pmatrix}0 & -1\\-1 & 0\end{pmatrix}

Rotation 90° clockwise about the origin

We can represent a rotation of 90° clockwise about the origin using the matrix:

\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}0&1\\-1&0\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}0&1\\-1&0\end{pmatrix}
• So:
  • A = \begin{pmatrix}0&1\\-1&0\end{pmatrix}
• Answer: \begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}

Rotation 90° anticlockwise about the origin

We can represent a rotation of 90° anticlockwise about the origin using the matrix:

\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}0&-1\\1&0\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}0&-1\\1&0\end{pmatrix}
• So:
  • A = \begin{pmatrix}0&-1\\1&0\end{pmatrix}
• Answer: \begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}

Rotation 180° about the origin

We can represent a rotation of 180° about the origin using the matrix:

\begin{pmatrix}-1 & 0\\0 & -1\end{pmatrix}

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}-1&0\\0&-1\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}-1&0\\0&-1\end{pmatrix}
• So:
  • A = \begin{pmatrix}-1&0\\0&-1\end{pmatrix}
• Answer: \begin{pmatrix}-1 & 0\\0 & -1\end{pmatrix}

Enlargement about the origin

We can represent an enlargement about the origin with scale factor k using the matrix:

\begin{pmatrix}k & 0\\0 & k\end{pmatrix}

…where k is the scale factor.

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}k&0\\0&k\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}k&0\\0&k\end{pmatrix}
• So:
  • A = \begin{pmatrix}k&0\\0&k\end{pmatrix}
• Answer: \begin{pmatrix}k & 0\\0 & k\end{pmatrix}

Horizontal stretch parallel to the x-axis

We can represent a horizontal stretch parallel to the x-axis with scale factor k using the matrix:

\begin{pmatrix}k & 0\\0 & 1\end{pmatrix}

…where k is the scale factor.

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}k&0\\0&1\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}k&0\\0&1\end{pmatrix}
• So:
  • A = \begin{pmatrix}k&0\\0&1\end{pmatrix}
• Answer: \begin{pmatrix}k & 0\\0 & 1\end{pmatrix}

Vertical stretch parallel to the y-axis

We can represent a vertical stretch parallel to the y-axis with scale factor k using the matrix:

\begin{pmatrix}1 & 0\\0 & k\end{pmatrix}

…where k is the scale factor.

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
  • \begin{pmatrix}1&0\\0&1\end{pmatrix}
• Write out where the points will end up, as a column matrix:
  • \begin{pmatrix}1&0\\0&k\end{pmatrix}
• Find the matrix A such that:
  • A \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}1&0\\0&k\end{pmatrix}
• So:
  • A = \begin{pmatrix}1&0\\0&k\end{pmatrix}
• Answer: \begin{pmatrix}1 & 0\\0 & k\end{pmatrix}

flashcards

Question	Answer
Reflection in the x-axis matrix	\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}
Reflection in the y-axis matrix	\begin{pmatrix}-1 & 0\\0 & 1\end{pmatrix}
Reflection in the line y=x matrix	\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}
Reflection in the line y=-x matrix	\begin{pmatrix}0 & -1\\-1 & 0\end{pmatrix}
Rotation 90° clockwise about origin matrix	\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}
Rotation 90° anticlockwise about origin matrix	\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}
Rotation 180° about origin matrix	\begin{pmatrix}-1 & 0\\0 & -1\end{pmatrix}
Enlargement about origin with scale factor k matrix	\begin{pmatrix}k & 0\\0 & k\end{pmatrix}
Horizontal stretch parallel to x-axis with scale factor k matrix	\begin{pmatrix}k & 0\\0 & 1\end{pmatrix}
Vertical stretch parallel to y-axis with scale factor k matrix	\begin{pmatrix}1 & 0\\0 & k\end{pmatrix}
Method to find transformation matrix from unit square points	Use coordinates of (1,0) and (0,1) as column matrix \begin{pmatrix}1&0\\0&1\end{pmatrix}, find where they map to as column matrix, then solve A \times \text{original} = \text{image} to get A
What does a reflection in the x-axis do to the point (x,y)?	Maps it to (x, -y)
What does a reflection in the y-axis do to the point (x,y)?	Maps it to (-x, y)
What does a reflection in y=x do to the point (x,y)?	Maps it to (y, x)
What does a reflection in y=-x do to the point (x,y)?	Maps it to (-y, -x)
What does a 90° clockwise rotation do to (x,y)?	Maps it to (y, -x)
What does a 90° anticlockwise rotation do to (x,y)?	Maps it to (-y, x)
What does a 180° rotation do to (x,y)?	Maps it to (-x, -y)
What does an enlargement scale factor k about origin do to (x,y)?	Maps it to (kx, ky)
What does a horizontal stretch scale factor k do to (x,y)?	Maps it to (kx, y)
What does a vertical stretch scale factor k do to (x,y)?	Maps it to (x, ky)
Unit square initial column matrix for finding transformations	\begin{pmatrix}1&0\\0&1\end{pmatrix}