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Chapter 5: Q4E (page 245)

Let Abe ann脳mmatrix. Is the formula(kerA)鈯�=im(AT)necessarily true? Explain.

Short Answer

Expert verified

Yes.

Step by step solution

01

Determine the property of orthogonal.

Consider an脳m matrix A.

Theorem: For any matrix B, imB鈯�=kerBT .

By the definition, imAT鈯�=kerATT.

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

imAT鈯�=kerATTimAT鈯�=kerAimAT鈯�鈯�=kerA鈯�imAT=kerA鈯�

Hence, the statement is true

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Most popular questions from this chapter

For which value(s) of the constant k are the vectors

u鈫�=[234]andv鈫�[1k1]and perpendicular?

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

15.[21-2]

If Ais ann脳mmatrix, is the formulaim(A)=im(AAT)necessarily true? Explain.

If A and B are arbitraryn脳n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

A螕BA.

Consider a symmetric n脳mmatrix A. What is the relationship between Im(A)and ker(A)?
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