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Unformatted text preview: d y be two vectors. Define the vector z = x + y where is a real (scalar) number. Find the value of that minimizes the norm of z. [3pts] Answer: This was one of the level C problems. Recall that to calculate norm of z, we must first calculate z.z. So, we start from here. z.z ( x y ).( x y ) x.x 2 x. y 2 y Now, we must take square root of z.z and find that minimizes it. This looks complicated since square root is not going to be easy. Now an easy but key observation is that the that maximizes square root of z.z will also maximize z.z. So, we can try to minimize z.z. This is easy! It is a quadratic function of . We can take derivative with respect of and set it to zero: d
( z.z ) 2 x. y 2 y. y 0 d If y is zero, then z=x and all values of lead to the same value for norm of z, i.e., ||x||. On the other hand, if y is non‐zero, we can divide by y.y and solve for : x. y y. y It will be instructive and fun for you to understand this answer geometrically by drawing vectors x, y, and z in the plane and interpreting norm as the length of the vector. Notice the close connection to component of x along y idea. 5. Find the partial fraction expansion of G ( s) ( s 2) [4pts] s ( s 1)( s 3) Answer: We start by writing the general form of the partial fraction expansion: A
C s s 1 s 3
A sG ( s ) |s 0 2 / 3 G ( s) B ( s 1)G ( s ) |s 1 1/ 2 C ( s 3)G ( s ) |s 3 1/ 6
G ( s) 2
1 3s 2( s 1) 6( s 3) 6. Consider two complex numbers w a jb
z re j a. Suppose a=1, b=2. It is known that z is complex conjugate of w. Find r and . [3pts] b. Now suppose a, b, r and are unknown real numbers. Suppose we are told that zw is purely imaginary and z2w is purely real. Find value(s) of . [3pts] Answer: For...
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This note was uploaded on 12/27/2011 for the course EEL 3105 taught by Professor Boykins during the Fall '10 term at University of Florida.
- Fall '10