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: Sylvester’s Inequality. In class, we’ve 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 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at Berkeley.
- Fall '10
- Electrical Engineering