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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

To make a trend analysis of six evenly spaced data points, one can use orthogonal polynomials with respect to evaluation at the points \(t = - 5, - 3, - 1,\,\,1,\,\,3,{\rm{ and }}5\).

  1. Show that the first three orthogonal polynomials are

\({p_0}\left( t \right) = 1,\,\,\,\,\,\,{p_1}\left( t \right) = t,{\rm{ and }}{p_2}\left( t \right) = \frac{3}{8}{t^2} - \frac{{35}}{8}\)

(The polynomial \({p_2}\) has been scaled so that its values at the evaluation points are small integers.)

  1. Fit a quadratic trend function to the data \(\left( { - 5,1} \right),\left( { - 3,1} \right),\left( { - 1,4} \right),\left( {1,4} \right),\left( {3,6} \right),\left( {5,8} \right)\).

Find an orthonormal basis of the kernel of the matrix A=[11111234].

(a) Consider an matrix A such that A螕A=Im. It is necessarily true that? Explain.

(b) Consider an n脳nmatrix A such that ATA=In. Is it necessarily true that AAT=In? Explain.

Consider an n x m matrix A with rank (A) = m. Is it always possible to write A as A = QL where Q is an n x m matrix with orthonormal columns and L is a lower triangular m x m matrix with positive diagonal entries? Explain.

Consider the linear systemAx鈫�=b鈫� , where

A=[1326]and b鈫�=[1020].

a. Draw a sketch showing the following subsets of 鈩�2:

  • The kernel ofA , and(kerA)鈯�
  • The image of AT
  • The solution setSof the system Ax鈫�=b鈫�

b.What relationship do you observe between(kerA) and im(AT)? Explain.

c. What relationship do you observe betweenrole="math" localid="1660916844921" ker(A) and S? Explain.

d. Find the unique vectorx鈫�0 in the intersection ofS and(kerA)鈯� . Show x鈫�0on your sketch.

e. What can you say about the length of x鈫�0compared with the length of all other vectors in S?
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