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Tag Archives: projection matrix
Linear Algebra and Its Applications, Exercise 3.3.11
Exercise 3.3.11. Suppose that is a subspace with orthogonal complement , with a projection matrix onto and a projection matrix onto . What are and ? Also, show that is its own inverse. Answer: Given any vector we have where … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.9
Exercise 3.3.9. Suppose that is a matrix such that . a) Show that is a projection matrix. b) If then what is the subspace onto which projects? Answer: a) To show that is a projection matrix we must show that and also … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.8
Exercise 3.3.8. Suppose that is a projection matrix from onto a subspace with dimension . What is the column space of ? What is its rank? Answer: Suppose that is a arbitrary vector in . From the definition of we know … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.7
Exercise 3.3.7. Given the two vectors and find the projection matrix that projects onto the subspace spanned by and . Answer: The subspace spanned by and is the column space where The projection matrix onto the subspace is then . We … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.6
Exercise 3.3.6. Given the matrix and vector defined as follows find the projection of onto the column space of . Decompose the vector into the sum of two orthogonal vectors and where is in the column space. Which subspace is in? … Continue reading
Posted in linear algebra
Tagged column space, left nullspace, orthogonal subspaces, orthogonal vectors, projection matrix
6 Comments
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 3.2.14
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