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 di±erent blocks? Problem 7: 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 ²nite 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...
View
Full Document
 Fall '10
 ClaireTomlin
 Electrical Engineering, Linear Algebra, Vector Space, Linear map

Click to edit the document details