dimension r − q
The nullity is unknown but we know it is > 0. This assumes that the dimension of the null space is equivalent to the rank r. However, if q = 0, so that \(t - \lambda_1\) is the zero map. But if \(t - \lambda_1\) is the 0 map, then the rank is the empty set, but the the nullity > 0. This is a contradiction. This proves that it cannot be the zero map.
The exact number of y's is not essential for the proof because they all go to \(\overrightarrow{0}\) (see end of proof).