 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-i/3/b/i/solution

Since

S^{'}

is the restriction of

S

im T

im (ST) = {S (T u) : u \in U} = {S (v) : v \in im T} = im S^{'} .

This image of a linear map is contained in

im S

, so

r_{ST} \leq r_{S}

. Also, applying

S

to a basis of

im T

im S^{'}

r_{T}

. Thus

r_{ST} \leq r_{T}

, and

r_{ST} \leq min (r_{S}, r_{T}) .

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