 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-i/2/b/iii/solution

No. The rank-nullity theorem gives

dim ker T_{A} = m - rank T_{A} \geq m - n > 0,

but this guarantees infinitely many solutions only when the linear system is consistent. For a counterexample take

n = 1, m = 2, A = (00), b = (1) .

There is no solution, since the left side always vanishes.

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