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Category Archives: linear algebra
Linear Algebra and Its Applications, Exercise 3.1.22
Exercise 3.1.22. Consider the equation and the subspace of containing all vectors that satisfy it. Find a basis for , the orthogonal complement of . Answer: is the nullspace of the linear system where Since is the nullspace of its … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.21
Exercise 3.1.21. If is the plane in described by what is the equation for the plane parallel to through the origin? What is a vector perpendicular to ? Find a matrix for which is the nullspace, and a matrix for … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.20
Exercise 3.1.20. Suppose is a subspace of . Show that . What does this mean? Answer: We first consider the case where ; in other words, contains only the zero vector. From exercise 3.1.18 we know that . The only … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.19
Exercise 3.1.19. State whether each of the following is true or false: (a) If the subspaces and are orthogonal, then and are also orthogonal. (b) If is orthogonal to and orthogonal to then is orthogonal to . Answer: (a) In … Continue reading
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
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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
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