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Linear Algebra With Applications
Found in: Page 245
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Let A be an n×m matrix. Is the formula (kerA)=im(AT) necessarily true? Explain.


See the step by step solution

Step by Step Solution

Step 1: Determine the property of orthogonal.

Consider a n×m matrix A.

Theorem: For any matrix B, imB=kerBT .

By the definition, imAT=kerATT.

As role="math" localid="1659498837555" ATT=Aand A=A , take both side in the equation imAT=kerATTas follows.

imAT=kerATT imAT=kerAimAT=kerA imAT=kerA

Hence, the statement is true

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