 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/ia/paper-1/8b/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-07

Construct the matrix of a linear map by expressing the images of the basis vectors in that same basis:

L e_{j} = \sum_{i = 1}^{2} A_{ij} e_{i} .

Thus the

j

th column is

[L e_{j}]_{B}

. For

v = v_{1} e_{1} + v_{2} e_{2}

[Lv]_{B} = A [v]_{B} .

The column

[Lv]_{B}

is a coordinate vector; reconstructing the actual vector requires

Lv = (A [v]_{B})_{1} e_{1} + (A [v]_{B})_{2} e_{2}

. This distinguishes the linear map from its basis-dependent matrix.

 New to topics? Read the docs here!