Matrices : Theory & Applications Solutions to the exercises
نویسنده
چکیده
(b) Let assume first that B be onto. Then apply the first question to A and BC ; we obtain rkA+rk(BC) ≤ m+rk(ABC). This is the desired inequality, since rkB = m and rk(AB) = dimR(AB) = dimR(A) = rkA. We now consider the general case. The previous analysis is obviously valid for homomorphisms. We apply it to the triplet (A′, B, C), where A′ is the restriction of A to R(B). We obtain rk(A′B) + rk(BC) ≤ rkB + rk(A′BC).
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