Linear transformations Cheat Sheet - Invariant points, matrices, determinants, and successive transformations

Oct 13, 2026

A concise cheat sheet explaining linear transformations as mappings of points to their images, emphasizing that only linear terms are involved and the origin maps to itself. It shows how to represent transformations with matrices, determine an unknown transformation using its effect on unit vectors, and interpret common transformations such as reflections, rotations, stretches, and enlargements. It also covers invariant points and lines, the determinant as the scale factor for area change, and how to combine transformations via matrix multiplication. Finally, it extends the ideas to three-dimensional linear transformations using unit vectors and corresponding matrices.

LanguageEnglish

CategoryPresentations

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Pages1

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Properties of linear transformations

Matrix representation and unit vectors

Invariant points and invariant lines

Common 2D transformations

Reflections, rotations, stretches, enlargements

Determinants and change in area

Successive transformations using matrices

Linear transformations in three dimensions

Linear transformations Cheat Sheet A linear transformation describes how a general point is transformed. The new point is called the image. Properties A linear transformation only involves linear terms in 􀝔 and 􀝕 . Below are three different transformations. The only one that is a linear transformation is 􀜶 since the transformation matrix has all entries written in the form 􀜽􀝔 + 􀜾􀝕 . 􀡿: (􀢞􀢟) ↦ (􀢞􀢟 􀫚􀫡) 􀢀: (􀢞􀢟) ↦ (􀫝􀢞􀢟􀢟) 􀢁: (􀢞􀢟) ↦ (􀢟􀢞􀫛 ) ▪ Any linear transformation can be represented by a matrix ▪ Linear transformations always map the origin onto itself. ▪ The linear transformation 􀜶: (􀝔􀝕) ↦ (􀜽􀜿􀝔􀝔 􀜾􀝀􀝕􀝕) can be represented by the matrix 􀜯 = (􀜽􀜿 􀜾􀝀) . Any linear transformation can be defined by its effect on the unit vectors (10) and (01) . You can use the following fact to find the transformation that a given matrix represents. ▪ If 􀜯 = (􀜽􀜿 􀜾􀝀), then (􀜽􀜿 􀜾􀝀) (10) = (􀜽􀜿) and (􀜽􀜿 􀜾􀝀) (01) = (􀜾􀝀) . | Example 1: 􀜯 = (10 1)
Describe fully the transformation represented by 􀜯 . | | | --- | --- | | We find what happens to the unit vectors under 􀜯: | (10 1) (10) = (10) , (10 1) (01) = (1) | | Drawing a diagram to visualise the effect 􀜯 had on the unit vectors: | | | The unit vectors have been reflected in the 􀝔 − 􀜽􀝔􀝅􀝏 . | (1,0) is unchanged by 􀜯 while (0, 1) is transformed to (0, −1).
We can conclude that 􀜯 represents a reflection in the 􀝔 −axis. | Reflections, rotations, enlargements and stretches ▪ Points which are mapped to themselves under a given transformation are known as invariant points. ▪ Lines which are mapped to themselves under a given transformation are known as invariant lines. The below table details the different matrices that correspond to particular transformations you need to be familiar with, as well as their respective invariant points and lines. You will be given the rotation matrix in the formula booklet, as well as the matrix for a reflection in the line 􀝕 = (􀝐􀜽􀝊􀟠)􀝔 . | Transformation
Reflection in the 􀝕 −axis
Reflection in the 􀝔 −axis Reflection in the line 􀝕 = 􀝔
Reflection in the line 􀝕 = −􀝔
Rotation through angle 􀟠 anticlockwise about the origin
Stretch of scale factor 􀜽 parallel to the 􀝔 −axis and a stretch of scale factor 􀜾 parallel to the 􀝕 −axis If 􀢇 = 􀢈, the transformation is an enlargement with scale factor 􀢇 Stretch of scale factor 􀜽 parallel to the 􀝔 −axis only
Stretch of scale factor 􀜾 parallel to the 􀝕 −axis only | Matrix
(1 01)(10 1)(01 10)(1 1)
(􀜿􀝏􀝋􀝅􀝏􀝊􀟠􀟠 􀝏􀝋􀝅􀝊􀟠􀝏􀟠 )
(􀜽0 0􀜾) | | Invariant points
All points on the
􀝕 −axis All points on the
􀝔 −axis All points on the
line 􀝕 = 􀝔 All points on the line 􀝕 = −􀝔 Only the origin (0, 0)
Only the origin
(0, 0)
All points on the
􀝕 −axis All points on the
􀝔 −axis | Invariant lines
􀝔 = 0 and 􀝕 = 􀝇 for any 􀝇
􀝕 = 0 and 􀝔 = 􀝇 for any 􀝇
The lines 􀝕 = −􀝔 and 􀝕 =−􀝔 + 􀝇 for any 􀝇 The lines 􀝕 = −􀝔 and 􀝕 =􀝔 + 􀝇 for any 􀝇 For 􀟠 ≠ 180° there are no invariant lines. For 􀟠 = 180° any line passing through the origin isan invariant line. The 􀝔 and 􀝕 axes.
If a = b, all lines passing through the origin, with any gradient, are invariant lines.
Any line parallel to the
􀝔 −axis Any line parallel to the
􀝕 −axis | | --- | --- | --- | --- | --- | | | (􀜽0
(10 | 01) 0􀜾) | | | Remember that you can always use the method from example 1 to find the transformation represented by a particular matrix if you are unsure of what it represents. The determinant ▪ If 􀜯 = (􀜽􀜿 􀜾􀝀) represents a linear transformation, then 􀝀􀝁􀝐􀜯 represents the scale factor for the change in area. The following formula is useful for questions involving the change in area as a result of a linear transformation. ▪ 􀜣􀝎􀝁􀜽􀯜􀯠􀯔􀯚􀯘 = 􀜣􀝎􀝁􀜽􀯢􀯕􀯝􀯘􀯖􀯧 × |􀝀􀝁􀝐􀜯| | Example 2: 􀜯 = (20 3)
(a) Describe fully the transformation represented by 􀜯 .
A 2D shape with area 􀝇 is transformed using the transformation represented by 􀜯 .
(b) Given that the image of the shape has area 24, find the value of 􀝇 . | | | --

FAQs

What defines a linear transformation and what basic property does it always satisfy?

A linear transformation uses only linear terms and maps the origin onto itself. The image of a point is called the transformed image.

How can a matrix representation of a transformation be identified using unit vectors?

Apply the transformation matrix to the unit vectors (1,0) and (0,1). The resulting columns correspond to how the basis directions are mapped, letting you determine the transformation.

How does the determinant relate to the area change under a 2D linear transformation?

If the transformation matrix is represented by 􀜯, then the determinant gives the area scale factor through the formula for area change: new area equals old area multiplied by the absolute value of the determinant.

Oct 13, 2026

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