Matrix transformations about the origin

Reflection in the x-axis

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

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer:

Reflection in the y-axis

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

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer:

Reflection in the line y = x

We can represent a reflection in the line using the matrix:

Finding this out

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

Reflection in the line y = -x

We can represent a reflection in the line using the matrix:

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer:

Rotation 90° clockwise about the origin

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

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer:

Rotation 90° anticlockwise about the origin

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

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer:

Rotation 180° about the origin

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

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer:

Enlargement about the origin

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

…where is the scale factor.

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer:

Horizontal stretch parallel to the x-axis

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

…where is the scale factor.

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer:

Vertical stretch parallel to the y-axis

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

…where is the scale factor.

Finding this out

• Write out the coordinates of two non-origin points on the unit square, as a column matrix:
• Write out where the points will end up, as a column matrix:
• Find the matrix such that:
• So:
• Answer: