49 math25-q.docx - Please email [email protected] for the answer to the below question You can also email us for help on any of your

# 49 math25-q.docx - Please email...

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Please email [email protected] for the answer to the below question. You can also email us for help on any of your assignments, testbanks etc. ECE1653: Hybrid Systems and Control Applications Homework 3 March 16, 2016 Due Date: April 4, 2016 1. Let { w 1 ,...,w r } be a linearly independent set of vectors in R n . Suppose we are given integers p and q such that 1 ≤ q p r . Define the subspaces of R n : V 1 = sp { w 1 ,...,w p } V 2 = Show that V 1 ∩ V 2 = sp { w q ,...,w p }. 2. Prove the following lemma. sp { w q ,...,w r } . Lemma 1. Suppose (R1)-(R4) hold. Also suppose that B∩C( v r 1 ) sp { b 1 ,...,b r 1 −1 } , where sp { b 1 ,...,b r 1 −1 } is the unique minimal subspace containing B ∩ C( v r 1 ) and generated by the linearly independent vectors { b 1 ,...,b r 1 −1 ,b r 1 +1 ,...,b m +1 } . There exists b r 1 B ∩ C( v r 1 ) b such that b r 1 = c 1 b 1 + ··· + c r 1 −1 b r 1 −1 , c i = 06 . Hint: First, argue by contradiction that B ∩ C( v r 1 ) contains vectors β i , i = 1 ,...,r 1 − 1, such that , with d i i 6= 0. Second, try to form a linear combination of these β i

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