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Chapter 5: Q29E (page 263)

TRUE OR FALSE

The formulaker(A)=ker(ATA)holds for all matrices A.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Sufficient condition

The given statement is true. For the proof, let x鈭�ker(A). So,

Ax鈫�=0鈫�鈬�ATAx鈫�=0鈫�鈬�x鈫�鈭�kerATA

02

Necessary condition

Conversely,

x鈭�kerATA鈬�ATAx鈫�=0鈫�鈬�x鈫�TATAx鈫�=0鈬�(Ax鈫�)T(Ax鈫�)=0鈬�Ax鈫�=0鈬�x鈫�鈭�ker(A)

Thus,ker(A)=ker(ATA)

Hence, the statement is true.

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