432hwk6 - x 1 + x 2 + x 3 = 3 x 3 = 1 x 1 + x 3 = 2 2 x 1 +...

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Math 432/532 Homework 15 points Due March 2, 2012 1. Find the equation of the line that gives the best “least-squares fit” to the data points ( - 2 , 12) , ( - 1 , 11) , (0 , 8) , (1 , 5) , (2 , 2) , (3 , - 3). 2. Find the point in the set { ( x 1 ,x 2 ,x 3 ) | x 1 + 3 x 2 - 2 x 3 = 0 , - 2 x 1 - 5 x 2 + 3 x 3 = 0 } that is nearest to the point (3 , 2 , 1). 3. Find the vector v R 3 of the form v = α (1 , 1 , 2) + β (2 , - 1 , 1) that is closest to (1 , 1 , 1). 4. Find the point in the plane x + 2 y + 3 z = 6 that is closest to the origin in R 3 . 5. Find the least squares solution of the linear system
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Unformatted text preview: x 1 + x 2 + x 3 = 3 x 3 = 1 x 1 + x 3 = 2 2 x 1 + 5 x 3 = 8-7 x 1 + 8 x 2 = 0 x 1 + 2 x 2-x 3 = 1 . 6. Find the minimum norm solution of the system 2 x 1 + x 2 + x 3 + 5 x 4 = 8-x 1-x 2 + 3 x 3 + 2 x 4 = 0 . 7. Let A be a matrix with linearly independent columns, and let M = R ( A ). Prove the following: (a) AA A = A (b) A A = ( A A ) (c) P M is symmetric. (d) P M 2 = P M ....
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