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Monthly Archives: March 2014
Linear Algebra and Its Applications, Exercise 3.1.18
Exercise 3.1.18. Suppose that is the subspace of containing only the origin. What is the orthogonal complement of ()? What is if is the subspace of spanned by the vector ? Answer: Every vector is orthogonal to the zero vector. … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.17
Exercise 3.1.17. Suppose that and are subspaces of and are orthogonal complements. Is there a matrix such that the row space of is and the nullspace of is ? If so, show how to construct using the basis vectors for … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.16
Exercise 3.1.16. Describe the set of all vectors orthogonal to the vectors and . Answer: If a vector is orthogonal to the vectors and then its inner products with those vectors must be zero, so that and This is a … 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 3.1.14
Exercise 3.1.14. Given two vectors and in , show that their difference is orthogonal to their sum if and only if their lengths and are the same. Answer: First we assume that is orthogonal to . This means that their … Continue reading
Posted in linear algebra
Tagged difference of vectors, orthogonal vectors, sum of vectors
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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 3.1.12
Exercise 3.1.12. For the matrix find a basis for the nullspace and show that it is orthogonal to the row space. Take the vector and express it as the sum of a nullspace component and a row space component . … Continue reading
Posted in linear algebra
Tagged basis, nullspace, nullspace component, row space component
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Linear Algebra and Its Applications, Exercise 3.1.11
Exercise 3.1.11. Fredholm’s alternative to the fundamental theorem of linear algebra states that for any matrix and vector either 1) has a solution or 2) has a solution, but not both. Show that assuming both (1) and (2) have solutions … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.10
Exercise 3.1.10. Given the two vectors and find a homogeneous system in three unknowns whose solutions are the linear combinations of the vectors. Answer: In the previous exercise 3.1.9 we showed that the plane spanned by the vectors and was … Continue reading
Posted in linear algebra
Tagged homogeneous system, linear combinations, three unknowns
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Linear Algebra and Its Applications, Exercise 3.1.9
Exercise 3.1.9. For the plane in spanned by the vectors and find the orthogonal complement (i.e., the line in perpendicular to the plane). Note that this can be done by solving the system where the two vectors are the rows … Continue reading