problem2 - and dim V = m be a linear map with rank( A ) = k...

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EE221A Linear System Theory Problem Set 2 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2010 Issued 9/10; Due 9/17 All answers must be justifed. Problem 1: Linearity. Let u and y be real scalar functions of time. Are the following maps H linear? (a) y ( t ) = H ( u ( t )) = u ( t ) (b) y ( t ) = H ( u ( t )) = i t 0 e - σ u ( t σ ) Problem 2: Linearity. Given A,B,C,X C n × n , determine if the following maps (involving matrix multi- plication) from C n × n C n × n are linear. 1. X m→ AX + XB 2. X m→ AX + BXC 3. X m→ AX + XBX Problem 3: Solutions to linear equations (this was part of Professor El Ghaoui’s prelim question this year). Consider the set S = { x : Ax = b } where A R m × n , b R m are given. What is the dimension of S ? Does it depend on b ? Problem 4: Rank-Nullity Theorem. Let A be a linear map from U to V with dim U = n and dim V = m . Show that dim R ( A ) + dim N ( A ) = n Problem 5: Representation of a Linear Map. Let A : ( U,F ) ( V,F ) with dim U = n
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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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