# rec3 - and v j m j =1 of U and V so that in these bases A...

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EE 221 : Linear Systems Recitation #3 — Sam Burden — Sep 10, 2010 1 Deﬁnition A metric space ( X,d ) is a set X with a metric d : X × X R for which d ( x,y ) 0 , d ( x,y ) = 0 ⇐⇒ x = y, d ( x,y ) = d ( y,x ) , d ( x,z ) d ( x,y ) + d ( y,z ) . Deﬁnition f : ( X,d ) ( Y,e ) is continuous if: ε > 0 : δ > 0 : d ( x,y ) < δ = e ( f ( x ) ,f ( y )) < ε. Exercise ( Hausdorﬀ ) For all x,y ( X,d ), there is a δ > 0 so that { w : d ( x,w ) < δ } \ { z : d ( y,z ) < δ } = . Fact L p ( I ) := ± f : I R | R I | f | p < ² is a (pseudo-)metric space with d ( f,g ) := (R I | f - g | p ) 1 /p . Exercise Consider A : ( L p ( I ) ,d ) ( R , |·| ) deﬁned by A ( f ) := f (0). (a) Is A linear? (b) Is A continuous? Cayley-Hamilton Deﬁnition The characteristic polynomial of A R n × n is χ ( s ) := det( sI - A ). Theorem ( Cayley-Hamilton ) χ ( A ) = 0. In particular there are α i so that A n = α 0 I + α 1 A + ··· + α n A n - 1 . Exercise Show that A n + p , p N , can be written as a linear combination of { A j } n - 1 0 . Bases Exercise Suppose A : ( U,F ) ( V,F ) with dim U = n and dim V = m is a linear map with rank A = k . Show that there are bases { u i } n i =1

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Unformatted text preview: and { v j } m j =1 of U and V so that in these bases A is represented by the block diagonal matrix A = ³ I 0 0 ´ . (a) What are the dimensions of the diﬀerent blocks? (b) Choose / construct the bases (and verify they are bases). (c) Show A has the right matrix representation. EE 221 : Linear Systems Recitation #3 — Sam Burden — Sep 10, 2010 2 Ax = B Exercise Let A ∈ C m × n , B ∈ C n × q , C ∈ C m × n , and D ∈ C n × q . (a) When is the matrix equation AX = C solvable for X ∈ C n × n ? When is it unique? (b) When is the matrix equation XB = D solvable for X ∈ C n × n ? When is it unique? (c) Under what conditions does TA = TC imply that A = C ?...
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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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rec3 - and v j m j =1 of U and V so that in these bases A...

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