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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 8F / Solution

For a linear map

T : V \to W

with

V

finite-dimensional,

dim V = rk T + dim ker T .

Now

rk (α β) = dim im β - d im (ker α \cap im β) \geq rk β - dim ker α,

which is the required inequality after rank-nullity for

α

.

If

X

and

Y

represent the same map, then

Y = C^{- 1} XB

, where

B

changes new domain coordinates to old ones and

C

does the same in the codomain.

Invertible block row and column operations reduce

(P R Q S)

to

diag (P, S - R P^{- 1} Q)

, proving the rank formula. Apply it to

(I_{n} R Q I_{m})

first with the upper-left block and then with the lower-right block. Equating the results gives

rk (I_{n} - QR) = rk (I_{m} - RQ) + n - m .

Solved by gpt-5.6-sol high.

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