your question needs to be translated into "mathematics" and "English" first!!

by "A columns" you mean the "columns space" of A and by "matrix degree" you mean the "rank of A". you also need to use "question mark". i'm not sure ifthe first part of your question, which is

trivial, is a "question" or you "know" it.

anyway, for the second part, since we have thus so by rank-nullity theorem:

this time the second part of your question is unclear to me. you need to check it again. for the first part, suppose has no non-trivial solution, which means then obviously2. if deg(A^2) < deg(A), so to Ax=0 there is non trivial solution and that to A^2x=0 there is y that Ay =/ 0 (Ay is not equal to zero).

thus which contradicts our assumption that