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Chapter 5: Q16E (page 263)

If Ais any matrix with ker(A)={0鈫�}then the matrix AAT represents the orthogonal projection onto the image of A.

Short Answer

Expert verified

False

Step by step solution

01

Theorem 

The matrix of an orthogonal projection: The matrix P of the orthogonal projection onto V=span{u鈫�1u鈫�2,......,u鈫�m}is given by, where Q={u鈫�1u鈫�2,......,u鈫�m} and QTis the transpose of Qand Q is an orthonormal basis of V.

02

Determine whether the statement is true or false

Thus, according to the matrix of an orthogonal projection.

A cannot be any matrix withker0=0. So that, A must be orthonormal.

Hence, the statement is false.

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Most popular questions from this chapter

TRUE OR FALSE?If A and Bare symmetric n脳nmatrices, then A+B must be symmetric as well.

Question: Consider an n脳mmatrix A, a vector in Rm, and a vector w鈫�in Rm. Show that (Av鈫�).w鈫�=v鈫�.(ATw鈫�).

Question:TRUE OR FALSE?If A and Bare symmetricn脳n matrices,AB then must be symmetric as well.

Are the rows of an orthogonal matrix A necessarily orthonormal?

Question: Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

9.[1111],[19-53]
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