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Category Archives: linear algebra
Linear Algebra and Its Applications, Exercise 3.3.3
Exercise 3.3.3. Given solve to find . Show that is orthogonal to every column in . Answer: We have or if is invertible. In this case we have so that We then have and The error vector is then The … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.2
Exercise 3.3.2. We have the value at time and the value at time . We wish to fit these values using a line constrained to go through the origin, i.e., with an equation of the form . Solve the system … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.1
Exercise 3.3.1. a) Given the system of equations consisting of and find the least squares solution and describe the error being minimized by that solution. Confirm that the error vector is orthogonal to the column . Answer: This is a … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.16
Exercise 3.2.16. a) Given the projection matrix projecting vectors onto the line through and two vectors and , show that the inner products of with and with are equal. b) In general would the angles between and and and be … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.15
Exercise 3.2.15. For a matrix show that if then the length of is equal to the length of for all . Answer: We have . By the rule for transposes of products we have so that . But since we … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.14
Exercise 3.2.14. In the planes corresponding to the equations and intersect in a line. What is the projection matrix that projects points in onto that line? Answer: To find the line of intersection we solve the system of equations where … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.13
Exercise 3.2.13. For a projection matrix show that the sum of the diagonal entries of (the “trace” of ) always equals one. Answer: For the diagonal entries of are , , through so that the trace of is UPDATE: Corrected … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.12
Exercise 3.2.12. Find a projection matrix that projects every vector in onto the line described by the equation . Answer: One solution to the equation is . The projection matrix that projects vectors onto the line through is NOTE: This … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.11
Exercise 3.2.11. a) Given the line through the origin and find the matrix that projects onto this line, as well as the matrix that projects onto the line perpendicular to the original line. b) What is ? What is ? … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.10
Exercise 3.2.10. Discuss whether the projection matrix from exercise 3.2.9 is invertible or not. Explain your answer. Answer:The projection matrix Note that each column of the resulting matrix is equal to the original vector multiplied by a scalar factor equal … Continue reading
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