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Tag Archives: projection matrix
Linear Algebra and Its Applications, Exercise 3.4.18
Exercise 3.4.18. If is the projection matrix onto the column space of the matrix and , what is a simple formula for ? Answer: The projection matrix onto the column space of can be calculated as . Since the columns … Continue reading
Posted in linear algebra
Tagged orthogonalization, orthonormal vectors, projection matrix
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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 3.3.20
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 3.3.17
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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Linear Algebra and Its Applications, Exercise 3.3.13
Exercise 3.3.13. Using least squares, find the line that is the best fit to the following measurements: at at at at Also, given the matrix find the projection of onto the column space . Answer: Assuming that the line in … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.12
Exercise 3.3.12. Given the subspace spanned by the two vectors and find the following: a) a set of basis vectors for b) the matrix that projects onto c) the vector in that has the minimum distance to the vector in Answer: … Continue reading
Posted in linear algebra
Tagged basis, column space, left nullspace, orthogonal complement, projection matrix
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