Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 1 / 8B / a / i

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- Solution i

Solution

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}

, linearity gives

[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.

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