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Category Archives: linear algebra
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
Linear Algebra and Its Applications, Exercise 3.3.26
Exercise 3.3.26. A middle-aged man is stretched on a rack with various forces. Given the measurements of length (in feet) at forces (in tons), and assuming that Hooke’s Law applies, use least squares to find the man’s length when no … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.25
Exercise 3.3.25. Given the measurements at from the previous exercise, what would be the coefficient matrix , the unknown vector , and the data vector if we wish to fit the data using a parabola of the form ? Answer: We would … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.24
Exercise 3.3.24. Given the measurements at use least squares to find the line of best fit. Answer: This corresponds to a system of the form as follows: To find the least squares solution we multiply both sides by to create a system … Continue reading
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