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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 = I 0 0 . (a) What are the dimensions of the dierent 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 ?...
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online