# 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

## 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

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## 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

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## 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

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## 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