# 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

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## 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

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## 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