For any linear operator A acting in Space Kn will A.0 be always equal to 0?

Question

This question comes from the assertion that a trivial subspace consisting only of element zero {0}, will be an invariant subspace for all A.

This means: A.0 = 0 for any A so that the invariance holds true.

Does this have to necessarily hold true for any operator A to be linear?

Answer 1

Yes! This will always be true for a linear operator $A$. To see this:

$A(0) = A(0+0) = A(0)+A(0) = 2A(0)$, and the only way this can happen is if $A(0)=0$.