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: 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.

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