 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-i/4/c/i/solution

Write

r = (r_{1}, \dots, r_{n})^{T}

. The linear map represented by

A

has the form

A x = (r^{T} x) e_{i} .

The image of a linear map here is contained in the one-dimensional linear span of

e_{i}

, so its matrix rank is at most one. The rank-nullity theorem yields

dim ker A = n - rank A \geq n - 1.

Thus the solution space of

A x = 0

, namely the zero eigenspace when zero is an eigenvalue, contains at least

n - 1

r = 0

n

; otherwise it is

n - 1

. When

n = 1

the asserted lower bound is zero, even if zero is not an eigenvalue.

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