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problem2 - A = n Problem 6 Representation of a Linear Map...

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EE221A Linear System Theory Problem Set 2 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2011 Issued 9/8; Due 9/16 All answers must be justified. Problem 1: Linearity. Are the following maps A linear? (a) A ( u ( t )) = u ( t ) for u ( t ) a scalar function of time (b) How about y ( t ) = A ( u ( t )) = integraltext t 0 e - σ u ( t σ ) ? (c) How about the map A : as 2 + bs + c integraltext s 0 ( bt + a ) dt from the space of polynomials with real coefficients to itelf? Problem 2: Nullspace of linear maps. Consider a linear map A . Prove that N ( A ) is a subspace. Problem 3: 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 mapsto→ AX + XB 2. X mapsto→ AX + BXC 3. X mapsto→ AX + XBX Problem 4: Solutions to linear equations (this was part of Professor El Ghaoui’s prelim question last 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 5: 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
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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...
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