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Unformatted text preview: A ) = n Problem 6: Representation of a Linear Map. Let A : ( U,F ) ( V,F ) with dim U = n and dim V = m be a linear map with rank( A ) = k . Show that there exist bases ( u i ) n i =1 , and ( v j ) m j =1 of U,V respectively such that with respect to these bases A is represented by the block diagonal matrix A = b I B What are the dimensions of the dierent blocks? Problem 7: Sylvesters Inequality. In class, weve discussed the Range of a linear map, denoting the rank of the map as the dimension of its range. Since all linear maps between nite dimensional vector spaces can be represented as matrix multiplication, the rank of such a linear map is the same as the rank of its matrix representation. Given A R m n and B R n p show that rank( A ) + rank( B ) n rank AB min [rank( A ) , rank( B )] 1...
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.
 Fall '10
 ClaireTomlin
 Electrical Engineering

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