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Monthly Archives: December 2017
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 3.4.7
Exercise 3.4.7. Given where are orthonormal vectors, compute and show that Answer:We have so that since the transpose of a sum is equal to the sum of the transposes. The product of the sums can then be decomposed into two … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.6
Exercise 3.4.6. Given the matrix find entries for the third column such that is orthogonal. How much freedom do you have to choose the entries? Finally, verify that both the columns and rows are orthonormal. Answer: In order for to … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.5
Exercise 3.4.5. Given a unit vector and , prove that is orthogonal. What is when ? Answer: We have Since the matrix is orthogonal. If then so that NOTE: This continues a series of posts containing worked out exercises from … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.4
Exercise 3.4.4. Given two orthogonal matrices and , show that their product is also orthogonal. If represents rotation through the angle and represents rotation through the angle , what does represent? What trigonometric identities for and can be found in … 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
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