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Chapter 1: Q44E (page 1)

Consider an n脳 mmatrix A. Find

dim(imA)+dim(kerAT)in terms of mand n.

Short Answer

Expert verified

dimkerAT+dimimA=n

Step by step solution

01

A is an n × m matrix

Here it needs to find out dimimA+dimkerAT.

Observe that for any n脳 mmatrix A it has,

dimimA=rankA 鈥︹�� (1)

According to the rank nullity theorem,

dimkerA=dimimA 鈥︹�� (2)

ABT=BTAT for all m脳 pmatrices Aand for all p脳 nmatrices B.

rankA=rankAT 鈥︹�� (3)

Since, A is an n脳 mmatrix, so ATis m x n .

From the equation (2) and (3) it gives,

dimkerAT+rankAT=n鈬�dimkerAT=n-rankAT鈬�dimkerAT=n-rankA鈬�dimkerAT+rankA=nHence,ThevalueofdimkerAT+rankA=n.

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