Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 19J / d

Table of contents
- Solution d

Solution

Take

t \mapsto A

and

r \mapsto B

. Then

A^{2} = I

and

A B A^{- 1} = (10 - 1 1) = B^{- 1},

so these matrices define a representation. The line

C e_{1}

is fixed by both matrices and is a one-dimensional subrepresentation.

Any complementary line has a generator

e_{2} + c e_{1}

. But

B (e_{2} + c e_{1}) = e_{2} + (c + 1) e_{1},

which cannot lie on the same line. Thus no complement is

B

-invariant, and therefore no

G

-invariant complement exists.

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