Homework 11 - M341 H11(S Zhang 3.5-6,6.1-3 1 Find the...

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Unformatted text preview: M341 H11 (S. Zhang) 3.5-6,6.1-3. 1. Find the transition matrix from [ e 1 , e 2 ] to [ u 1 , u 2 ], then the transition matrix from [ u 1 , u 2 ] to [ e 1 , e 2 ]. u 1 = 1 1 , u 2 =- 1 1 ans: (3.5:1a,2a) A given vector can be written as linear combinations of different basis. v = v 1 e 1 + v 2 e 2 = [ e 1 , e 2 ] v 1 v 2 v = u 1 u 1 + u 2 u 2 = [ u 1 , u 2 ] u 1 u 2 u 1 u 2 = U- 1 I v 1 v 2 = S v 1 v 2 In general, from V to U , S = U- 1 V . From [ u 1 , u 2 ] to [ e 1 , e 2 ]. S = I- 1 U = U = 1- 1 1 1 From [ e 1 , e 2 ] to [ u 1 , u 2 ], S = U- 1 I = 1 / 2 1 / 2- 1 / 2 1 / 2 u1=[1 1]; u2=[-1 1] U=[u1 u2] iU=inv(U) 2. Find the transition matrix from [ e 1 , e 2 ] to [ u 1 , u 2 ], then the transition matrix from [ u 1 , u 2 ] to [ e 1 , e 2 ]. u 1 = 1 2 , u 2 = 2 5 ans: (3.5:1b,2b) A given vector can be written as linear combinations of different basis. v = v 1 e 1 + v 2 e 2 = [ e 1 , e 2 ] v 1 v 2 v = u 1 u 1 + u 2 u 2 = [ u 1 , u 2 ] u 1 u 2 u 1 u 2 = U- 1 I v 1 v 2 = S v 1 v 2 In general, from V to U , S = U- 1 V . From [ u 1 , u 2 ] to [ e 1 , e 2 ]. S = I- 1 U = U = 1 2 2 5 From [ e 1 , e 2 ] to [ u 1 , u 2 ], S = U- 1 I = 5- 2- 2 1 u1=[1 2]; u2=[2 5] U=[u1 u2] iU=inv(U) 3. Find the transition matrix from [ e 1 , e 2 ] to [ u 1 , u 2 ], then the transition matrix from [ u 1 , u 2 ] to [ e 1 , e 2 ]. u 1 = 1 2 , u 2 = 2 5 ans: (3.5:1c,2c) A given vector can be written as linear combinations of different basis. v = v 1 e 1 + v 2 e 2 = [ e 1 , e 2 ] v 1 v 2 v = u 1 u 1 + u 2 u 2 = [ u 1 , u 2 ] u 1 u 2 u 1 u 2 = U- 1 I v 1 v 2 = S v 1 v 2 In general, from V to U , S = U- 1 V . From [ u 1 , u 2 ] to [ e 1 , e 2 ]. S = I- 1 U = U = 1 1 From [ e 1 , e 2 ] to [ u 1 , u 2 ], S = U- 1 I = 1 1 u1=[0 1]; u2=[1 0]; U=[u1 u2] iU=inv(U) 4. Let u 1 = ( 1 , 1 , 1 ) T , u 2 = ( 1 , 2 , 2 ) T , u 3 = ( 2 , 3 , 4 ) T . Find the transition matrix corresponding to the change of basis from [ e 1 , e 2 , e 3 ] to [ u 1 , u 2 , u 3 ]. Find the coordinates of each of the following vectors with respect to [ u 1 , u 2 , u 3 ]. v 1 = 3 2 5 , v 2 = 1 1 2 , v 3 = 2 3 2 ans: (3.5:5) The transition matrix S = U- 1 I = 1 1 2 1 2 3 1 2 4 - 1 = 2- 1- 1 2- 1- 1 1 v 1 = 3 2 5 , v u = Sv v = S 3 2 5 = 1- 4 3 1 To check: Uv u = V v = 1 1 2 1 2 3 1 2 4 1- 4 3 = 3 2 5 v 2 = 1 1 2 , v u = Sv v = S 1 1 2 = - 1 1 v 2 = 2 3 2 , v u = Sv v = S 2 3 2 = 1 u1=[1 1 1]; u2=[1 2 2]; u3=[2 3 4]; U=[u1 u2 u3] iU=inv(U) iU*[3 2 5; 1 1 2;2 3 4] 5. Let u 1 = ( 1 , 1 , 1 ) T , u 2 = ( 1 , 2 , 2 ) T , u 3 = ( 2 , 3 , 4 ) T . Let v 1 = ( 4 , 6 , 7 ) T , v 2 = ( , 1 , 1 ) T , v 3 = ( , 1 , 2 ) T . Find the transition matrix from [ v 1 , v 2 , v 3 ] to [ u 1 , u 2 , u 3 ]. Find the coordinates of...
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Homework 11 - M341 H11(S Zhang 3.5-6,6.1-3 1 Find the...

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