recMT - Sam Burden Linear Systems EE 221 Midterm Review —...

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Unformatted text preview: Sam Burden Linear Systems ( EE 221 ) Midterm Review — Oct 15, 2010 1 Outline of Course 1. Preliminaries: f : X → Y , F , ( V,F ), { b j } n 1 (a) functions: injective, surjective, bijective, left & right inverse (b) field, ring (c) vector space, subspace (d) linear (in)dependence (e) basis, coordinate representation 2. Linear Maps: A : V → W , R ( A ), N ( A ), A ∈ F n × m (a) range & null spaces (b) matrix representation (c) change-of-basis, similarity transform (d) Sylvester’s inequality 3. Normed & Inner Product Spaces: |·| , h· , ·i , A * , A = U Σ V * (a) norm equivalence (b) induced norms (c) inner products (d) adjoints (e) orthogonality, Gram-Schmidt (f) Projection Theorem, least-squares (g) SVD 4. Differential Equations: ˙ x = f ( x,t ), s ( t,t ,x ,u ), ρ ( t,t ,x ,u ), Φ( t,τ ) (a) piecewise, Lipschitz continuity (b) Cauchy sequence, Banach space (c) Bellman-Gronwall Lemma (d) Fundamental Theorem (e) dynamical system representation (f) state transition and output readout maps (g) time invariance 5. System Representation R ( · ) = ( A ( · ) ,B ( · ) ,C ( · ) ,D ( · )), ˙ ξ ≈ D x fξ + D u fμ (a) state transition matrix (b) linearization Sam Burden Linear Systems ( EE 221 ) Midterm Review — Oct 15, 2010 2 Past Recitations L P spaces Definition The Lebesgue space L P ( I ) := n f : I → R | (R I | f ( x ) | p dx ) 1 /p < ∞ o . Exercise Naturally a vector space under the operations ( f + g )( x ) := f ( x ) + g ( x ) , ( λf )( x ) := λf ( x ) . Question L p ⊂ L q ? Linear Independence Definition { x α } α ∈ I is linearly independent if each finite subcollection is linearly indepen- dent. Exercise Show that { e t ,e 2 t } is linearly independent in ( C ([0 , 1]) , R ). Continuity & Linearity Definition A metric space ( X,d ) is a set X with a metric d : X × X → R for which d ( x,y ) ≥ , 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 Consider A : ( L p ( I ) ,d ) → ( R , |·| ) defined by A ( f ) := f (0). (a) Is A linear? (b) Is A continuous? 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 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 different blocks? (b) Choose / construct the bases (and verify they are bases). (c) Show A has the right matrix representation. Sam Burden Linear Systems ( EE 221 ) Midterm Review — Oct 15, 2010 3 Ax = B Exercise Let A ∈ C m × n , B ∈ C n × q , C ∈ C m × n , and D ∈ C n × q ....
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recMT - Sam Burden Linear Systems EE 221 Midterm Review —...

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