All subjects · Further Mathematics · WFM01 Further Pure Mathematics 1

Linear transformations and matrices

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

A 2 × 2 matrix acts on points of the plane: multiply the matrix by the position vector of a point to get its image. Transformations that can be done this way are called linear.

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Images of points

The image of the point (x, y) under M = (a b; c d) is M times the column vector (x, y): (ax + by, cx + dy).

A linear transformation always maps O to O, and maps straight lines to straight lines.

The columns tell you everything

M(1, 0) = (a, c) and M(0, 1) = (b, d). So the first column is the image of (1, 0) and the second column is the image of (0, 1).

To find the matrix of a transformation, find where it sends (1, 0) and (0, 1) and write those images as columns.

Transforming a shape

Write the vertices as columns of one matrix and multiply: M(x_1 x_2 x_3; y_1 y_2 y_3) gives all the images at once. The unit square with vertices O, (1, 0), (1, 1), (0, 1) is the usual test shape.

Worked example

Find the image of the point (2, 5) under T = (3 1; -1 2).

  1. (3×2 + 1×5, -1×2 + 2×5)

Answer: (11, 8)

Key idea: Image of (x, y) = M(x, y). The columns of M are the images of (1, 0) and (0, 1).

Check yourself

  1. Find the x-coordinate of the image of (2, 5) under (3 1; -1 2).
    Show answer

    11. 3×2 + 1×5 = 11.

  2. Find the y-coordinate of the image of (2, 5) under (3 1; -1 2).
    Show answer

    8. -1×2 + 2×5 = 8.

  3. A linear transformation maps (1, 0) to (2, 3) and (0, 1) to (-1, 4). What is its matrix?
    • (2 3; -1 4)
    • (2 -1; 3 4)
    • (-1 2; 4 3)
    • (3 2; 4 -1)
    Show answer

    (2 -1; 3 4). The images are the columns: first column (2, 3), second column (-1, 4).

  4. The unit square is transformed by (3 1; -1 2). Find the x-coordinate of the image of the vertex (1, 1).
    Show answer

    4. 3×1 + 1×1 = 4. (The image of (1, 1) is the sum of the two columns.)

  5. Which transformation cannot be represented by a 2 × 2 matrix?
    • Translation by (3, 1)
    • Rotation 90° about O
    • Reflection in y = x
    • Enlargement scale factor 2 about O
    Show answer

    Translation by (3, 1). Any 2 × 2 matrix maps O to O, but a translation moves O.

  6. T = (3 1; -1 2) maps the point (p, 0) to (12, -4). Find p.
    Show answer

    4. T(p, 0) = (3p, -p) = (12, -4), so p = 4.

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Last updated 8 November 2026. Spotted a mistake? Tell us.