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Tag Archives: projection
Linear Algebra and Its Applications, Exercise 3.4.21
Exercise 3.4.21. Given the function on the interval , what is the closest function to ? What is the closest line to ? Answer: To find the closest function to the function we first project onto the function on the … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.10
Exercise 3.4.10. Given the two orthonormal vectors and and an arbitrary vector , what linear combination of and is the least distance from ? Show that the difference between and that combination (i.e., the error vector) is orthogonal to both … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.9
Exercise 3.4.9. Given the three orthonormal vectors , , and , what linear combination of and is the least distance from ? Answer: Any linear combination of and is in the plane formed by and . The combination closest to … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.8
Exercise 3.4.8. Project the vector onto the two non-orthogonal vectors and and show that the sum of the two projections does not equal (as it would if and were orthogonal). Answer: The projection of onto is . We have and … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.3
Exercise 3.4.3. Given the orthonormal vectors and and the vector from the previous exercise, project onto a third orthonormal vector . What is the sum of the three projections? Why? Why is the matrix equal to the identity matrix ? … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.2
Exercise 3.4.2. Given two orthonormal vectors and and the vector , project onto and . Also find the projection of onto the plane formed by and . Answer: Since is orthonormal, the projection of onto is given by Similarly the … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.1
Exercise 3.4.1. a) Given the following four data points: write down the four equations for fitting to the data. b) Find the line fit by least squares and calculate the error . c) Given the value of what is in … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.5
Exercise 3.3.5. Given the system with no solution, provide a graph of a straight line that minimizes and solve for the equation of the line. What is the result of projecting the vector onto the column space of ? Answer: … Continue reading
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Tagged column space, error vector, least squares, projection
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Linear Algebra and Its Applications, Exercise 3.3.4
Exercise 3.3.4. Given expand the expression , compute its partial derivatives with respect to and , and set them to zero. Compare the resulting equations to to confirm that you obtain the same normal equations in both cases (i.e., using … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.3
Exercise 3.2.3. Find the multiple of the vector that is closest to the point . Also find the point on the line through that is closest to . Answer: The first problem amounts to finding the projection of onto . … Continue reading
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