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quiz2_1806_s05 - By recursion or cofactors or otherwise...

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1 2 3 4 5 6 18.06 Professor Strang Quiz 2 April 1, 2005 Grading Your PRINTED name is:
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1 (17 pts.) If the output vectors from Gram-Schmidt are cos θ sin θ q 1 = and q 2 = sin θ cos θ describ e all possible input vectors a 1 and a 2 . 2
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2 (15 pts.) If a and b are nonzero vectors in R n , what number x minimizes the squared length b xa 2 ? 3
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3 (17 pts.) Find the projection p of the vector b = (1 , 2 , 6) onto the plane x + y + z = 0 in R 3 . (You may want to find a basis for this 2-dimensional subspace, even an orthogonal basis.) 4
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4 (17 pts.) Find the determinants of A and A 1 and the (1 , 2) entry of A 1 if A = 0 0 1 0 1 1 0 0 1 2 1 3 1 3 1 7 . 5
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5 (17 pts.) By recursion or cofactors or otherwise(!) compute the determinant of this 5 by 5 circulant matrix C : 2
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Unformatted text preview: By recursion or cofactors or otherwise(!) compute the determinant of this 5 by 5 circulant matrix C : ⎡ ⎤ 2 − 1 0 − 1 ⎢ ⎥ ⎢ ⎥ ⎢ − 1 2 − 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ C = ⎢ 0 − 1 2 − 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 − 1 2 − 1 ⎥ ⎣ ⎦ − 1 0 − 1 2 6 6 (17 pts.) Suppose P 1 is the projection matrix onto the 1-dimensional subspace spanned by the first column of A . Suppose P 2 is the projection matrix onto the 2­ dimensional column space of A . After thinking a little, compute the product P 2 P 1 . ⎡ ⎤ 1 0 ⎢ ⎥ ⎢ ⎥ ⎢ 2 1 ⎥ ⎢ ⎥ A = . ⎢ ⎥ ⎢ 0 1 ⎥ ⎣ ⎦ 1 2 7...
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