aem570-w10-midterm - d 1 m,n = d m,n[1 d m,n 3 Let M denote...

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AA/EE/ME 570: Manifolds and Geometry for Systems and Control Midterm Exam Due: Thursday February 18, 8:00pm The exam is open book, open note. You may use as much time as you like to complete it up until 8:00pm on February 18. There are five problems each worth 10 points for a total of 50 points. 1. Let V be an F -vector space with U V a subspace and U 0 V a complement to U in V (i.e., V = U U 0 ). Show that there is a natural isomorphism between U 0 and V/U . 2. Let M be a set. A metric on M is a function d : M × M R such that for all m 1 , m 2 , m 3 M d ( m 1 , m 2 ) = 0 iff m 1 = m 2 ) d ( m 1 , m 2 ) = d ( m 2 , m 1 ) d ( m 1 , m 3 ) d ( m 1 , m 2 ) + d ( m 2 , m 3 ) . The diameter of a set is given by diam ( M ) = sup { d ( u, v ) | u, v M } . Show that every metric space has an equivalent metric in which the diameter of the space is 1. Hint: Consider the new metric
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Unformatted text preview: d 1 ( m,n ) = d ( m,n ) / [1 + d ( m,n )] . 3. Let M denote the set of continuous functions f : [0 , 1] → R on the interval [0 , 1] . Show that d ( f,g ) = Z 1 | f ( x )-g ( x ) | dx is a metric. 4. Let E = F = R n with the standard norms and A,B ∈ L ( E ; F ) . Let < A,B > = trace ( AB T ) . Show that this choice is an inner product on L ( E ; F ) . 5. Let t = 2 e 1 ⊗ e 1-e 2 ⊗ e 1 + 3 e 1 ⊗ e 2 and φ ∈ L ( R 2 ; R 2 ) , ψ ∈ L ( R 3 ; R 2 ) be given by the matrices ± 2 1-1 1 ² , ± 0 1-1 1 0 2 ² . Compute: trace( t ), φ * t , ψ * t , trace ( φ * t ) , trace ( ψ * t ) , φ * t . 1...
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