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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...
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