Tag Archives: column space

Linear Algebra and Its Applications, Exercise 3.4.15

Posted on December 28, 2017 by hecker

Exercise 3.4.15. Given the matrix find the orthonormal vectors and that span the column space of . Next find the vector that completes the orthonormal set, and describe the subspace of of which is an element. Finally, for find the … Continue reading →

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Linear Algebra and Its Applications, Exercise 3.4.1

Posted on December 14, 2017 by hecker

Exercise 3.4.1. a) Given the following four data points: write down the four equations for fitting to the data. b) Find the line fit by least squares and calculate the error . c) Given the value of what is in … Continue reading →

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Linear Algebra and Its Applications, Exercise 3.3.14

Posted on October 22, 2016 by hecker

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 →

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Linear Algebra and Its Applications, Exercise 3.3.12

Posted on October 8, 2016 by hecker

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 →

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Linear Algebra and Its Applications, Exercise 3.3.8

Posted on September 10, 2016 by hecker

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 →

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Linear Algebra and Its Applications, Exercise 3.3.7

Posted on September 3, 2016 by hecker

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 →

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Linear Algebra and Its Applications, Exercise 3.3.6

Posted on July 10, 2016 by hecker

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 3.3.5

Posted on October 25, 2014 by hecker

Exercise 3.3.5. Given the system with no solution, provide a graph of a straight line that minimizes and solve for the equation of the line. What is the result of projecting the vector onto the column space of ? Answer: … Continue reading →

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Linear Algebra and Its Applications, Exercise 3.1.13

Posted on March 15, 2014 by hecker

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 →

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Linear Algebra and Its Applications, Exercise 3.1.7

Posted on February 22, 2014 by hecker

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 →

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