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homework3_solutions - MATH 211(Siefken Homework#3 Due...

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Unformatted text preview: MATH 211 (Siefken): Homework #3 Due June 16, 2011 Question #1 (a) Let A = 1- 1- 1 1 1 1 1 1 . (a) Find a maximal linearly independent set of mutually orthogonal vectors that are also orthogonal to the rows of A . That is, come up with a linearly independent set S of vectors that has the properties: All vectors in S are orthogonal to the rows of A , all vectors in S are orthogonal to eachother, and if another vector is added to S , either S becomes linearly dependent or S violates one of the previous two rules. Let r 1 and r 2 be the rows of A . From Homework 1, we know that s 1 = 1- 1 1- 1 and s 2 = 1 1- 1- 1 are orthogonal to r 1 , r 2 and in fact { r 1 , r 2 , s 1 , s 2 } is a maximal set of mutually orthogonal vectors. The set is maximal since it spans R 4 , and so any other vector would be linearly dependent on r 1 , r 2 , s 1 , s 2 . We may therefore construct the set S = { s 1 , s 2 } and we know S has the desired properties. (b) How do row( A ), null( A ), and span( S ) relate to eachother? Since null( A ) is the set of all vectors orthogonal to R ( A ) and S is a maximal set of vectors orthogonal to the rows of A , we know null( A ) = span( S ). Finally, we know all vectors in R ( A ) are orthogonal to all vectors in both null( A ) and span( S ). Question #2 Suppose A x = 3 2 7 has complete solution x = 1 + 1 1 s + - 1 1 t . (a) Find rank( A ). We know that the complete solution to A x = b is of the form p + null( A ) where p is a particular solution....
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homework3_solutions - MATH 211(Siefken Homework#3 Due...

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