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m526t3_s10_pract

# m526t3_s10_pract - C Dt to ﬁt b = 1 2 1 4 3 5 at times t...

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Math 526 Practice Test 3 April 13, 2010 pr.1 Let A be the matrix 2 6 - 4 4 10 30 - 20 6 8 24 - 16 8 . (a) Find a basis for the subspace Null ( A ), Null ( A T ), Col ( A ), Col ( A T ); Find the dimension of each of the four subspaces. (b) Show that the subspaces Null ( A T ) and Col ( A ) are orthogonal. (c) Show that the subspaces Null ( A ) and Col ( A T ) are orthogonal. pr.2 Let V be the linear space whose basis are the vectors and v 1 = 1 0 1 1 and v 2 = 0 1 3 1 . Let W be the linear space whose basis are the vectors w 1 = 1 3 - 1 0 and w 2 = 1 1 0 - 1 . (a) Prove that V and W are orthogonal spaces. (b) What is the dimension of V ? What is the dimension of W ? Is W the orthogonal complement of V ? Explain. pr.3 Consider the space V whose basis are the two vectors 1 - 1 1 , and 3 2 2 . Let b = 4 1 3 . Find the projection of the vector b onto V . pr.4 Find the best line

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Unformatted text preview: C + Dt to ﬁt b = 1 , , 2 , 1 , 4 , 3 , 5 at times t =-3 ,-2 ,-1 , , 1 , 2 , 3. pr.5 Compute the determinant of the matrix A = 7 9 5 6 2 11 0 3 8 13 7 7 . pr.6 For each of the following answer True or False. (a) If A is a square matrix with determinant 2010, then the dimension of Null ( A T ) is zero. (b) If the vector v is orthogonal to itself, that is v · v = , then v = . (c) If A is 7 × 9 matrix with rank 5, then dim ( Null ( A T )) = 4. (d) The projection of a vector onto a subspace V is always a vector in V . (e) If the vectors a and b are orthogonal, and the vectors b and c are orthogonal, then the vectors a and c are orthogonal....
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m526t3_s10_pract - C Dt to ﬁt b = 1 2 1 4 3 5 at times t...

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