This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 ,..., r of M * M . Denote by u j the columns of U and by v j the columns of V , where M * Mv j = j v j for j = 1 ,...,r . Prove the following (you may wish to revisit the proof of the SVD, and to use properties stated in problem 3.): (i) The span of v r +1 ,v r +2 ,... is Ker( M ) (=the nullspace of M ). (ii) The span of u 1 ,...,u r is Ran( M ) (=the column space of M ). (iii) The span of u r +1 ,u r +2 ,... is Ker( M * ) (=the left nullspace of M ) (iv) The span of v 1 ,...,v r is Ran( M * ) (=the row space of M ). 5. Prove the following properties of the pseudoinverse M + of the matrix M (you may wish to use properties listed in problem 4.): (i) MM + is the orthogonal projector on Ran( M ). (ii) M + M is the orthogonal projector on Ker( M ) . 2...
View Full
Document
 Winter '08
 un
 Linear Algebra, Algebra, Matrices

Click to edit the document details