Proof: If TS = 0, then (TS)^2 = 0

To prove that (TS)^2 = 0, given that TS = 0, we will show that for any vector v in V, ((TS)^2)(v) = 0.

Let v be any vector in V. We have ((TS)^2)(v) = (TS)(TS)(v).

Since TS = 0, we can rewrite the expression as ((TS)^2)(v) = (0)(TS)(v).

Since any linear map multiplied by the zero map is also the zero map, we have ((TS)^2)(v) = 0.

Therefore, we have shown that (TS)^2 = 0, as ((TS)^2)(v) = 0 for all v in V.