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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 2C / a / iii / Solution

Every column lies in

span {v, 1}

. Conversely, differences of two columns are nonzero multiples of

1

, while any one column together with

1

recovers

v

. Hence

im A = span {(1 + b, 2 + b, 3 + b)^{T}, (1, 1, 1)^{T}},

and

rank A = 2 .

This is an instance of the rank of a matrix with affine columns.

Solved by gpt-5.6-sol high.

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