prac4 - MAS 3114Practice Problem Set #4 1. Let vectorx 1 =...

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Unformatted text preview: MAS 3114Practice Problem Set #4 1. Let vectorx 1 = (3 , 5 , 1 , 1) T and vectorx 2 = (- 1 , 1 ,- 1 ,- 1) T , and set W = Span { vectorx 1 ,vectorx 2 } . (a) Compute the distance between vectorx 1 and vectorx 2 . (b) Find an orthonormal basis { vectoru 1 ,vectoru 2 } for W . (c) Let vectorw = (0 , 4 ,- 1 ,- 1) T W . Express vectorw as a linear combination of vectoru 1 and vectoru 2 . (d) Let vector y = (1 ,- 1 , , 2) T . Find vectors y W and vector z W such that vector y = y + vector z . 2. Short answer questions. If there is not enough information given to answer the question, then say so. (a) Let W = Span { (1 , 2 , 3 , 4) T , (5 , 6 , 7 , 8) T } . Write the equation(s) that the entries of the vector vectorx = ( x 1 ,x 2 ,x 3 ,x 4 ) T R 4 must satisfy for vectorx to be in W . (b) Define what it means for the set { vectorx 1 ,vectorx 2 ,...,vectorx k } R n to be orthogonal. (c) What is the orthogonal complement of the null space of A T ? (Possible an- swers: the null space of A , the null space of A T , the row space of A , the column space of A .) (d) Give an orthonormal subset of R 3 which is as large as possible....
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prac4 - MAS 3114Practice Problem Set #4 1. Let vectorx 1 =...

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