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Unformatted text preview: 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 diFerent blocks? Problem 6: 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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- Fall '10
- Electrical Engineering