This preview shows page 1. Sign up to view the full content.
Unformatted text preview: to show that rank( A T A ) = rank( A ) and null( A T A ) = null( A ) . Problem 4. Consider the linear transformation T : R 3 R 3 dened by T ( ~x ) = M ~x in terms of the 3 3 matrix M = 2 9 13 3 3 2 6 20 11 . a. Compute the QRfactorization of M . b. Find an orthonormal basis for the image of T . Problem 5. Let V and W be linear spaces with inner products hi V and hi W , respectively. A linear transformation T : V W is said to be an isometry if it preserves lengths i.e.,  T ( f )  W =  f  V for all f V . a. Show that every isometry is a monomorphism i.e., if T is an isometry then ker( T ) = { } . b. Say V = W = R n , and that hi V = hi W is the dot product. Show that T is an isometry if and only if it is orthogonal....
View
Full
Document
This note was uploaded on 03/31/2012 for the course MA 351 taught by Professor ?? during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 ??
 Linear Algebra, Algebra

Click to edit the document details