What does each matrix do to the unit square?

The columns of a 2×2 matrix are the images of and . Pick a transformation, watch the green image of the unit square, and read off the matrix. These are the OCR-spec transformations.

Identity & rotations about (0,0)
Reflections
Stretches, enlargement & shears (drag k)
Working: transform all four corners at once

Write the four corners as the columns of a single matrix, then multiply by . Each column of the answer is one transformed corner; join back up to draw the green shape.

A and B are the unit points on the axes, so columns and are exactly the two columns of .

Formula Sheet (OCR)

The OCR formula book gives only the two general results below. Everything in the second group is not in the book – you read it off the columns or get it from the general formula. = the area scale factor.

TransformationMatrix
Given in the OCR formula book
Rotation anticlockwise by about
Reflection in the line
Not given – read off the columns (or use the formulae above)
90° anticlockwise
180°
90° clockwise
Reflection in the -axis
Reflection in the -axis
Reflection in
Reflection in
Stretch parallel to (-axis invariant, s.f. )
Stretch parallel to (-axis invariant, s.f. )
Enlargement, centre (s.f. )
Shear parallel to (-axis fixed)
Shear parallel to (-axis fixed)

Rotation by any angle

Drag the angle. A(1,0) is the unit point on the x-axis and B(0,1) is the unit point on the y-axis. Rotating the whole plane just means rotating these two points: A swings to (cosθ, sinθ) and B to (−sinθ, cosθ). Those are the two columns of the matrix.

θ = 40°
Method – where do A and B go?
1
is the unit point on the x-axis. Rotate it by and it lands at – that is the first column of .
2
is the unit point on the y-axis. Rotate it by and it lands at – the second column.
3
Stack those two images as columns:
Why only A and B?
and are the two basic axis points. Every other point is built from them – for example . So once you know where A and B rotate to, every point follows by the same scaling (this is linearity). That is why fixing the images of A and B fixes the whole rotation, and why the matrix only needs those two columns.

Doing one transformation then another

If you apply first and then , the single matrix that does both is the product - the first one you apply goes on the right. Order usually matters.

First : then :
Is the same?

Swapping the order applies first instead. Usually this gives a different matrix – order matters.

Blue dashed = after .  Green = after both. The original unit square is grey.

Invariant points and lines

Invariant means never changing. Press Play below and watch every point slide from where it starts (black) to where the matrix sends it (red). The interesting question is: which points and lines don’t change?

Overview
Invariant point

A single point that maps to itself. When the transformation is applied it lands straight back in the same place, while the points around it move.

Invariant line

A line that maps onto itself. Points are allowed to slide along the line to a new spot, but they never leave it.

Line of invariant points

A whole line where every point stays put. Each point on the line maps exactly to itself; points off the line still move. (It is a special, “frozen” invariant line.)

Dynamic Examples

Load one into (or type your own), then press Play.

Two invariant lines through the origin; points slide along them.
The origin is the only fixed point; nothing else stays.
A whole line of points is frozen in place.
A shear – tick the box below to see every horizontal line is invariant.
Matrix T
Invariant points
Invariant lines
Line of invariant points
Method 1 – finding invariant points
1
An invariant point maps to itself, so and . Multiply out and set each component equal to the original:
2
Take and across so each line equals zero:
3
Solve the two equations together. always works. If the two equations are multiples of each other they describe one line, so every point on it is invariant – a line of invariant points. Otherwise the origin is the only one.

(In matrix form this is the same as , if you prefer that route.)

Worked numerical example
1
Take . Set :
2
Rearrange each to equal zero:
3
They are the same equation, so there is a whole line of invariant points: .
4
Check : – it maps to itself. ✓
Method 2 – finding invariant lines
1
Try a line . A general point on it is .
2
Find its image under :
3
The image must lie on the same line for every , i.e. . Matching the -terms gives the gradient equation:
4
Each real root is the gradient of an invariant line through . Matching the constants gives , so off the origin you also need – then every parallel line of that gradient is invariant.

(Quicker version for lines through : just require the image of to keep gradient – it gives the same quadratic.)

Worked numerical example
1
Take , so .
2
Substitute into :
3
Factorise: , so or .
4
Check , direction : – still gradient 3. ✓

Transformations in 3D

A 3×3 matrix transforms 3D space. Its three columns are the images of i = (1,0,0), j = (0,1,0) and k = (0,0,1). Pick a transformation and watch the green image of the unit cube. = the volume scale factor.

Reflection in a plane
Rotation about an axis (drag θ)
Other
unit cube image i′ j′ k′
Standard 3D matrices
TransformationMatrix
Reflection in a coordinate plane
In (flip )
In (flip )
In (flip )
Rotation anticlockwise by about an axis
About the -axis
About the -axis
About the -axis
Enlargement
Centre , s.f. (volume )