# Tag Archives: row space

## 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 3.1.15

Exercise 3.1.15. Is there a matrix such that the vector is in the row space of the matrix and the vector is in the nullspace of the matrix? Answer: The row space of any matrix is the orthogonal complement to … Continue reading

## Linear Algebra and Its Applications, Exercise 3.1.13

Exercise 3.1.13. Provide a picture showing the action of in sending the column space of to the row space and the left nullspace to zero. Answer: I’m leaving this post as a placeholder until I have time to illustrate this. … Continue reading

## Linear Algebra and Its Applications, Exercise 3.1.7

Exercise 3.1.7. For the matrix find vectors and such that is orthogonal to the row space of and is orthogonal to the column space of > Answer: The nullspace of is orthogonal to the row space of . We can … Continue reading

## Linear Algebra and Its Applications, Review Exercise 2.33

Review exercise 2.33. Consider the following factorization: a) What is the rank of ? b) Find a basis for the row space of . c) Are rows 1, 2, and 3 of linearly independent: true or false? d) Find a … Continue reading

## Linear Algebra and Its Applications, Review Exercise 2.8

Review exercise 2.8. Do the following: a) Find a matrix whose nullspace contains the vector . b) Find a matrix whose left nullspace contains the vector . c) Find a matrix with column space spanned by and row space spanned … Continue reading