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Category Archives: linear algebra
Linear Algebra and Its Applications, Exercise 3.3.23
Exercise 3.3.23. Given measurements show that the best least squares fit to the horizontal line is given by Answer: This corresponds to the system where is an by 1 matrix with all entries equal to 1 and . To find the … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.22
Exercise 3.3.22. Given measurements of at use least squares to find the line of best fit of the form . Answer: This corresponds to a system as follows: To find the least squares solution we form the system . We have and … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.21
Exercise 3.3.21. Given three vectors , , and , and the two lines through the origin and and through in the direction of , we want to find scalar values and such that the distance between the points and is … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.20
Exercise 3.3.20. Given the matrix that projects onto the row space of , find the matrix that projects onto the nullspace of . Answer: The null space of is orthogonal to the row space of . The two spaces are orthogonal complements, with … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.19
Exercise 3.3.19. Given a matrix , the matrix projects onto the column space of . Find the matrix that projects onto the row space of . Answer: The row space of is the column space of . We can then … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.18
Exercise 3.3.18. Suppose we have the following measurements of : and want to fit a plane of the form . a) Write a system of 4 equations in 3 unknowns representing the problem. (The system may not have a solution.) b) Write … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.17
Exercise 3.3.17. Find the projection matrix that projects vectors in onto the line . Answer: The vector is a basis for the subspace being projected onto, which is thus the column space of Using the formula we have so that and … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.16
Exercise 3.3.16. Suppose is a vector with unit length. Show that the matrix (with rank 1) is a projection matrix. Answer: We have But since has unit length we have so that We also have Since and the rank-1 matrix is … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.15
Exercise 3.3.15. Suppose is a projection matrix that projects vectors onto a line in the – plane. Describe the effect of the reflection matrix geometrically. Why does ? (Give both a geometric and algebraic explanation.) Answer: When applied to a … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.14
Exercise 3.3.14. Find the projection matrix onto the plane spanned by the vectors and . Find a nonzero vector that projects to zero. Answer: The plane in question is the column space of the matrix The projection matrix . We have … Continue reading
Posted in linear algebra
Tagged column space, left nullspace, plane, projection matrix
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