IBMathsDPApplications & Interpretation (AI)HLRevision Notes3. Geometry & TrigonometryMatrix TransformationsMatrix Transformations

Matrix Transformations (DP IB Applications & Interpretation (AI)): Revision Note

Paul

Written by: Paul

Reviewed by: Dan Finlay

Updated on

DP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HL

Transformation by a matrix

What is a transformation matrix?

  • A transformation maps an object to its image

    • e.g. translations, rotations, reflections, stretches, enlargements, etc

  • Matrices can be used to describe the transformation

    • They are called transformation matrices

  • In this course, the matrix transformations will be of the form

open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses plus open parentheses table row e row f end table close parentheses

  • The matrix open parentheses table row a b row c d end table close parentheses is normally labelled bold italic T

How do I find the coordinates of an image after a transformation?

  • STEP 1
    Write the coordinates of the vertices of the original object as column vectors

    • e.g. (0, 1), (0, 2) and (3, 1) are written as open parentheses table row 0 row 1 end table close parentheses, open parentheses table row 0 row 2 end table close parentheses and open parentheses table row 3 row 1 end table close parentheses

  • STEP 2
    Substitute each column vector into the matrix transformation

    • e.g. using open parentheses table row 1 0 row cell negative 3 end cell 1 end table close parentheses open parentheses table row x row y end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses

      • open parentheses table row 1 0 row cell negative 3 end cell 1 end table close parentheses open parentheses table row 0 row 1 end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses equals open parentheses table row 1 row 2 end table close parentheses

      • open parentheses table row 1 0 row cell negative 3 end cell 1 end table close parentheses open parentheses table row 0 row 2 end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses equals open parentheses table row 1 row 3 end table close parentheses

      • open parentheses table row 1 0 row cell negative 3 end cell 1 end table close parentheses open parentheses table row 3 row 1 end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses equals open parentheses table row 4 row cell negative 7 end cell end table close parentheses

  • STEP 3
    Write the column vectors as coordinates

    • e.g. (0,1) ⇾ (1, 2), (0,2) ⇾ (1, 3) and (3,1) ⇾ (4, -7)

Examiner Tips and Tricks

You can use your GDC. You can even do multiple coordinates at once. Just use a matrix where each column is a different pair of coordinates. If you do this then you need to make the addition column vector match.

For example, you could do open parentheses table row 1 0 row cell negative 3 end cell 1 end table close parentheses open parentheses table row 0 0 3 row 1 2 1 end table close parentheses plus open parentheses table row 1 1 1 row 1 1 1 end table close parentheses equals open parentheses table row 1 1 4 row 2 3 cell negative 7 end cell end table close parentheses.

The matrix of coordinates of the original object is usually labelled bold italic P. The matrix of coordinates of the image is usually labelled bold italic P apostrophe.

How do I find the coordinates of the original point given the image under a transformation?

  • You might be asked to find the original coordinates given

    • a matrix transformation bold italic T open parentheses table row x row y end table close parentheses plus open parentheses table row e row f end table close parentheses

    • and the coordinates of an image open parentheses x apostrophe comma space y apostrophe close parentheses

  • You just rearrange bold italic T open parentheses table row x row y end table close parentheses plus open parentheses table row e row f end table close parentheses equals open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses to find open parentheses x comma space y close parentheses

  • For example, given open parentheses table row 1 0 row cell negative 3 end cell 1 end table close parentheses open parentheses table row x row y end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses equals open parentheses table row 3 row 0 end table close parentheses

    • open parentheses table row 1 0 row cell negative 3 end cell 1 end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row 2 row cell negative 1 end cell end table close parentheses

    • open parentheses table row x row y end table close parentheses equals open parentheses table row 1 0 row 3 1 end table close parentheses open parentheses table row 2 row cell negative 1 end cell end table close parentheses equals open parentheses table row 2 row 5 end table close parentheses

    • (2, 5) ⇾ (3, 0)

Worked Example

A quadrilateral, Q, has the four vertices A(2, 5), B(5, 9), C(11, 9) and D(8, 5).

Find the coordinates of the image of Q under the transformation bold italic T equals open parentheses table row 3 cell negative 1 end cell row cell negative 1 end cell 2 end table close parentheses.

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    Paul

    Author: Paul

    Expertise: Maths Content Creator

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.